tag:blogger.com,1999:blog-46774856448613770012024-02-07T09:24:48.015-08:00MATH WITH YEASIR [mental math & IQ]Let's surprise everyone with our math and IQ talent!Unknownnoreply@blogger.comBlogger16125tag:blogger.com,1999:blog-4677485644861377001.post-72237933856794965192013-06-05T07:15:00.002-07:002013-06-05T07:22:27.522-07:00Divisibility Rules<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2nK60fHXZ5sdcAn9P93KIeRcHgdQBJDXdcCMop9phllY5_F9IgfASr5KWJu7RkQvj3iPR1s0cvu_S5SB4He6RQfa-KuzK6CS-ik3SLHdB2wzd8z0obLPgOAxKMm3xuSu2d2SDvYeYqVyJ/s1600/944405_518012974921591_1197754295_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2nK60fHXZ5sdcAn9P93KIeRcHgdQBJDXdcCMop9phllY5_F9IgfASr5KWJu7RkQvj3iPR1s0cvu_S5SB4He6RQfa-KuzK6CS-ik3SLHdB2wzd8z0obLPgOAxKMm3xuSu2d2SDvYeYqVyJ/s200/944405_518012974921591_1197754295_n.jpg" width="200" /></a></div>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 3</b>
<br />
<ul>Add up the digits: if the sum is divisible by three, then the number is as well.
<b>Examples</b>:
<ol>
<li>111111: the digits add to 6 so the whole number is divisible by three. </li>
<li>87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three.
</li>
</ol>
<b>Why does the 'divisibility by 3' rule work?</b></ul>
<pre>From: Dr. Math
To: Kevin Gallagher
Subject: Re: Divisibility of a number by 3
As Kevin Gallagher wrote to Dr. Math
On 5/11/96 at 21:35:40 (Eastern Time),
>I'm looking for a SIMPLE way to explain to several very bright 2nd
>graders why the divisibility by 3 rule works, i.e. add up all the
>digits; if the sum is evenly divisible by 3, then the number is as well.
>Thanks!
>Kevin Gallagher
The only way that I can think of to explain this would be as follows:
Look at a 2 digit number:
10a + b = 9a + (a + b).
We know that 9a is divisible by 3, so 10a + b will be divisible by 3
if and only if a + b is. Similarly,
100a + 10b + c = 99a + 9b + (a + b + c),
and 99a + 9b is divisible by 3, so the total will be iff a + b + c is.
This explanation also works to prove the divisibility by 9 test.
It clearly originates from modular arithmetic ideas, and I'm not sure if
it's simple enough, but it's the only explanation I can think of.</pre>
<ul><br />
<blockquote>
<pre>
</pre>
</blockquote>
Another visitor suggests this as an easier explanation, relying on decomposition and place value:
<br />
<blockquote>
<pre>We know that 9 is divisible by 3
10 = 9 + 1
and
30 = 9*3 + 3
Similarly,
3000 = 999*3 + 3
With this kind of decomposition in mind, examine any number;
for example, 1235:
1235 = 1000
200
30
+ 5
Now,
1000 = 999*1 + 1
200 = 99*2 + 2
30 = 9*3 + 3
5 = 5
Add these remainders:
1 + 2 + 3 + 5 = 11
Eleven is not divisible by three, which tells us that
our initial number, 1235, is not divisible by 3.
Another interesting fact about 3 is that any 3-digit
number with sequential digits, e.g., 123, 234, 456,
is divisible by 3.
</pre>
</blockquote>
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 4</b>
<br />
<ul>Look at the last two digits. If the number formed by its last two digits is divisible by 4, the original number is as well. <br /><b>Examples</b>:
<ol>
<li>100 is divisible by 4.
</li>
<li>1732782989264864826421834612 is divisible by four also, because 12 is divisible by four.
</li>
</ol>
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 5</b>
<br />
<ul>If the last digit is a five or a zero, then the number is divisible by 5.
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 6</b>
<br />
<ul>Check 3 and 2. If the number is divisible by both 3 and 2, it is divisible by 6 as well.
</ul>
<ul>Robert Rusher writes in:
<blockquote>
<pre>Another easy way to tell if a [multi-digit] number is divisible by six . . .
is to look at its [ones digit]: if it is even, and the sum of the [digits] is
a multiple of 3, then the number is divisible by 6.
</pre>
</blockquote>
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 7</b>
<br />
<ul>To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number. <br /><b>Example</b>: If you had 203, you would double the last digit to get six,
and subtract that from 20 to get 14. If you get an answer divisible by 7
(including zero), then the original number is divisible by seven. If you
don't know the new number's divisibility, you can apply the rule again.
</ul>
<ul>Matthew Correnti describes this method:
</ul>
<pre>If you do not know if a two-digit number, call it ab, is divisible
by 7, calculate 2a + 3b. This will yield a smaller number, and if
you do the process enough times you will eventually -- if the
number ab is divisible by 7 -- end up with 7.
You can use a similar method if you have a three-digit number abc:
take the digit a and multiply it by 2, then add it to the number bc,
giving you 2a + bc; repeat and reduce until you recognize the
result's divisibility by seven. With a four-digit number abcd, take
the digit a and multiply by 6, then add 6a to bcd giving. This
usually gives you a three-digit number; call it xyz. Take that x
and, as described previously, multiply x by two and add to yz
(i.e., 2x + yz). Again, repeat and reduce until you recognize the
result's divisibility by seven.</pre>
<ul><blockquote>
<pre></pre>
</blockquote>
</ul>
<ul>Ahmed Al Harthy writes in:
</ul>
<pre>To know if a number is a multiple of seven or not, we can use also
3 coefficients (1 , 2 , 3). We multiply the first number starting
from the ones place by 1, then the second from the right by 3,
the third by 2, the fourth by -1, the fifth by -3, the sixth by -2,
and the seventh by 1, and so forth.
Example: 348967129356876.
6 + 21 + 16 - 6 - 15 - 6 + 9 + 6 + 2 - 7 - 18 - 18 + 8 + 12 + 6 = 16
means the number is not multiple of seven.
If the number was 348967129356874, then the number is a multiple of seven
because instead of 16, we would find 14 as a result, which is a multiple of 7.
So the pattern is as follows: for a number onmlkjihgfedcba, calculate
a + 3b + 2c - d - 3e - 2f + g + 3h + 2i - j - 3k - 2l + m + 3n + 2o.
Example: 348967129356874.
Below each digit let me write its respective figure.
3 4 8 9 6 7 1 2 9 3 5 6 8 7 4
2 3 1 -2 -3 -1 2 3 1 -2 -3 -1 2 3 1
(3×2) + (4×3) + (8×1) + (9×-2) + (6×-3) + (7×-1) +
(1×2) + (2×3) + (9×1) + (3×-2) + (5×-3) + (6×-1) +
(8×2) + (7×3) + (4×1) = 14 -- a multiple of seven.</pre>
<ul><blockquote>
<pre></pre>
</blockquote>
</ul>
<ul>Another visitor observes:
</ul>
<pre>Here is one formula for seven...
3X + L
L = last digit
X = everything in front of last digit.
All numbers that are divisible by seven have this in common.
There are no exceptions.
For example, 42: 3(4) + 2 = 14.
Seven divides into 14, so it divides into 42.
Next example, 105: 3(10) + 5 = 35.
Seven divides into 35, so it divides into 105.
Here is another formula for seven:
4X - L
When using this formula, if you get zero, seven or a multiple of seven,
the number will be divisible by seven.
For example, 56: 4(5) - 6 = 14.
Seven divides into 14, so it divides into 56.
Next example, 168: 16(4) - 8 = 56.
Seven divides into 56, so it divides into 168.
Similarly:
The formula for 2 is 2X + L
The formula for 3 is 4X + L
The formula for 4 is 6X + L
The formula for 5 is 5X + L
The formula for 6 is 2X + L </pre>
<pre>and 4X + L -- in other words, the formulas for 2 and 3
must work before the number is divisible by 6.
The formula for 9 is X + L
The formula for 11 is X - L
The formula for 12 is 2X - L
The formula for 13 is 3X - L
The formula for 14 is 4X - L </pre>
<pre>and 2X + L -- in other words, the formulas for 7 and 2
must work before the number is divisible by 14.
The formula for 17 is 7X - L
The formula for 21 is X - 2L
The formula for 23 is 3X - 2L
The formula for 31 is X - 3L</pre>
<ul><blockquote>
<pre></pre>
</blockquote>
</ul>
<ul><a href="http://www.blogger.com/null" name="heikali7">Sara Heikali explains</a> this way to test a number with three or more digits for divisibility by seven:
<blockquote>
<pre>1. Write down just the digits in the tens and ones places.
2. Take the other numbers to the left of those last two digits,
and multiply them by two.
3. Add the answer from step two to the number from step one.
4. If the sum from step three is divisible by seven, then the
original number is divisible by seven, as well. If the sum is
not divisible by seven, then the original number is not
divisible by seven.
For example, if the number we are testing is 112, then
1. Write down just the digits in the tens and ones places: 12.
2. Take the other numbers to the left of those last two digits,
and multiply them by two: 1 × 2 = 2.
3. Add the answer from step two to the number from step one:
12 + 2 = 14.
4. Fourteen is divisible be seven. Therefore, our original
number, one hundred twelve, is also divisible by seven.
</pre>
</blockquote>
</ul>
<br />
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 8</b><br />
<ul>Check the last three digits. Since 1000 is divisible by 8, if the
last three digits of a number are divisible by 8, then so is the whole
number.<br /><b>Example</b>: <b>33333888</b> is divisible by 8; 33333886 isn't.
How can you tell whether the last three digits are divisible by 8? Phillip
McReynolds answers:
<br />
If the first digit is even, the number is divisible by 8 if the last two
digits are. If the first digit is odd, subtract 4 from the last two digits;
the number will be divisible by 8 if the resulting last two digits are. So,
to continue the last example,
<b>33333888</b> is divisible by 8 because the digit in the hundreds place is an
even number, and the last two digits are 88, which is divisible by 8.
33333886 is not divisible by 8 because the digit in the hundreds place is an
even number, but the last two digits are 86, which is not divisible by 8.
</ul>
<ul>
Sara Heikali explains this test of divisibility by eight for numbers with three or more digits:
<blockquote>
<pre>1. Write down the units digit of the original number.
2. Take the other numbers to the left of the last digit,
and multiply them by two.
3. Add the answer from step two to the number from step one.
4. If the sum from step three is divisible by eight, then the
original number is divisible by eight, as well. If the sum is
not divisible by eight, then the original number is not
divisible by eight.
For example, if the number we are testing is 104, then
1. Write down just the digits in ones place: 4.
2. Take the other numbers to the left of that last digit,
and multiply them by two: 10 × 2 = 20.
3. Add the answer from step two to the number from step one:
4 + 20 = 24.
4. Twenty-four is divisible be eight. Therefore, our original
number, one hundred and four, is also divisible by eight.
</pre>
</blockquote>
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 9</b>
<br />
<ul>Add the digits. If that sum is divisible by nine, then the original number is as well.
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 10</b>
<br />
<ul>If the number ends in 0, it is divisible by 10.
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 11</b>
<br />
<ul>Let's look at <b>352</b>, which is divisible by 11; the answer is 32. 3+2 is 5; another way to say this is that 35 -2 is 33.
Now look at <b>3531</b>, which is also divisible by 11. It is not a coincidence that 353-1 is 352 and 11 × 321 is 3531.
<br />
Here is a generalization of this system. Let's look at the number <b>94186565</b>.
<br />
First we want to find whether it is divisible by 11, but on the way we
are going to save the numbers that we use: in every step we will
subtract the last digit from the other digits, then save the subtracted
amount in order. Start with<br />
<pre> 9418656 - 5 = 9418651 SAVE 5
Then 941865 - 1 = 941864 SAVE 1
Then 94186 - 4 = 94182 SAVE 4
Then 9418 - 2 = 9416 SAVE 2
Then 941 - 6 = 935 SAVE 6
Then 93 - 5 = 88 SAVE 5
Then 8 - 8 = 0 SAVE 8</pre>
Now write the numbers we saved in reverse order, and we have <b>8562415</b>, which multiplied by 11 is <b>94186565</b>.
</ul>
<blockquote>
<blockquote>
<hr />
</blockquote>
</blockquote>
<ul>
Here's an even easier method, contributed by <b>Chis Foren</b>:
Take any number, such as <b>365167484</b>.<br />
Add the first, third, fifth, seventh,.., digits.....3 + 5 + 6 + 4 + 4 = <b>22</b>
<br />
Add the second, fourth, sixth, eighth,.., digits.....6 + 1 + 7 + 8 = <b>22</b>
<br />
If the difference, including 0, is divisible by 11,
then so is the number. <br />
<b>22 - 22 = 0</b> so <b>365167484</b> is evenly divisible by <b>11</b>.<br />
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 12</b>
<br />
<ul>
Check for divisibility by 3 and 4.
</ul>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 13</b>
<br />
<ul>
Here's a straightforward method supplied by Scott Fellows:
Delete the last digit from the given number. Then subtract nine times
the deleted digit from the remaining number. If what is left is
divisible by 13, then so is the original number.
</ul>
<a href="http://www.blogger.com/null" name="divide13">
</a>
<ul><blockquote>
<hr />
</blockquote>
Rafael Ando contributes:
Instead of deleting the last digit and subtracting it ninefold from the
remaining number (which works), you could also add the deleted digit
fourfold. Both methods work because 91 and 39 are each multiples of 13.
<br />
For any prime p (except 2 and 5), a rule of divisibility could be "created" using this method:
</ul>
<blockquote>
<blockquote>
<ol>
<li>Find m, such that m is the (preferably) smallest multiple of p that ends in either 1 or 9.
</li>
<li>Delete the last digit and add (if multiple ends in 9) or
subtract (if it ends in 1) the deleted digit times the integer nearest
to m/10. For example, if m = 91, the integer closest to 91/10 = 9.1 is
9; and for 3.9, it's 4.
</li>
<li>Verify if the result is a multiple of p. Use this process until it's obvious.
</li>
</ol>
Example 1: Let's see if 14281581 is a multiple of 17.
<br />
In this case, m = 51 (which is 17×3), so we'll be deleting the last number and subtracting it fivefold.
<br />
<br />
<blockquote>
1428158 - 5×1 = 1428153
<br />
142815 - 5×3 = 142800
<br />
14280 - 5×0 = 14280
<br />
1428 - 5×0 = 1428
<br />
142 - 5×8 = 102
<br />
10 - 5×2 = 0, which is a multiple of 17, so 14281581 is multiple of 17.
</blockquote>
Example 2: Let's see if 7183186 is a multiple of 46.
<br />
First, note that 46 is not a prime number, and its factorization is
2×23. So, 7183186 needs to be divisible by both 2 and 23. Since it's an
even number, it's obviously divisible by 2.
<br />
So let's verify that it is a multiple of 23:
<br />
<br />
<blockquote>
m = 3×23 = 69, which means we'll be adding the deleted digit sevenfold.
<br />
718318 + 7×6 = 718360
<br />
71836 + 7×0 = 71836
<br />
7183 + 7×6 = 7225
<br />
722 + 7×5 = 757
<br />
75 + 7×7 = 124
<br />
12 + 7×4 = 40
<br />
4 + 7×0 = 4 (not divisible by 23), so 7183186 is not divisible by 46.
</blockquote>
</blockquote>
Note that you could've stopped calculating whenever you find the result
to be obvious (i.e., you don't need to do it until the end). For
example, in example 1 if you recognize 102 as divisible by 17, you don't
need to continue (likewise, if you recognized 40 as not divisible by
23).
<br />
The idea behind this method it that you're either subtracting m×(last
digit) and then dividing by 10, or adding m×(last digit) and then
dividing by 10.
</blockquote>
<br />
<blockquote>
<blockquote>
<blockquote>
<a href="http://www.blogger.com/null" name="divide13">
</a>
<hr />
<a href="http://www.blogger.com/null" name="divide13">
</a></blockquote>
</blockquote>
<a href="http://www.blogger.com/null" name="divide13">
</a>
<ul>
Jeremy Lane adds:
<br />
It may be noted that while applying these rules, it is possible to loop among numbers as results.
<br />
Example: Is 1313 divisible by 13?
<br />
Using the procedure given we take 13×3 and obtain 39. This multiple ends in 9 so we add four-fold the last digit.
<br />
<blockquote>
131 + 4×3 = 143
<br />
14 + 4×3 = 26
<br />
2 + 4×6 = 26
<br />
...
</blockquote>
Example: Is 1326 divisible by 13?
<br />
Using the procedure given we take 13×7 = 91. This is not the smallest
multiple, but it does show looping. The smaller multiple does loop at 39
as well. There are some examples where we would still need to recognize
certain multiples. So we subtract nine-fold the last digit.
<br />
<blockquote>
132 - 9×6 = 78
<br />
7 - 9×8 = -65 (factor out -1)
<br />
6 - 9×5 = -39 (again factor out -1)
<br />
3 - 9×9 = -78 (factor out -1)
</blockquote>
This only occurs though if the number does happen to be divisible by the
prime divisor. Otherwise, eventually you will have a number that is
less than the prime divisor.
</ul>
<blockquote>
<blockquote>
<hr />
</blockquote>
</blockquote>
<ul>
And here's a more complex method that can be extended to other formulas:
<pre>1 = 1 (mod 13)
10 = -3 (mod 13) (i.e., 10 - -3 is divisible by 13)
100 = -4 (mod 13) (i.e., 100 - -4 is divisible by 13)
1000 = -1 (mod 13) (i.e., 1000 - -1 is divisible by 13)
10000 = 3 (mod 13)
100000 = 4 (mod 13)
1000000 = 1 (mod 13)
</pre>
Call the ones digit a, the tens digit b, the hundreds digit c, .....
and you get:
<ul>
a - 3×b - 4×c - d + 3×e + 4×f + g - .....
</ul>
If this number is divisible by 13, then so is the original number.
You can keep using this technique to get other formulas for divisibility for
prime numbers. For composite numbers just check for divisibility by
divisors.
</ul>
<a href="http://www.blogger.com/null" name="divide13">
</a>
<b><img src="http://mathforum.org/k12/mathtips/Tic.gif" /> Dividing by 14</b>
<br />
<ul>
Sara Heikali builds on her divisibility test for seven:
<pre>How can you know if a number with three or more digits
is divisible by the number fourteen?
Check if the last digit of the original number is odd or
even. If the number is odd, then the number is not
divisible by fourteen. If the number is even, then apply
the divisibility rule for seven.
(Keep in mind, the odd and even test is to see if the number
is divisible by two.) If the original even number is
divisible by seven, then it is also divisible by fourteen.
If the original even number is not divisible by seven, it
is not divisible by fourteen.
</pre>
</ul>
</blockquote>
</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-55046728768872197612013-06-05T07:00:00.001-07:002013-06-05T07:15:42.042-07:00Short-cut to find square of any number<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLcn6hBqgOe6e_Ozmxvz6PR4QB5wIeE7o2WPh1PSV8YOf8qqVnweI3x2kOOCnKTYYJ593hCoCdZFXLwzbENJtvIHOGQloGHgRG6RGw8eZRvFKzHQc9CmDQY5rx8aT5-rb0uWS9YS9ccnco/s1600/945520_448664931892903_1321992920_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLcn6hBqgOe6e_Ozmxvz6PR4QB5wIeE7o2WPh1PSV8YOf8qqVnweI3x2kOOCnKTYYJ593hCoCdZFXLwzbENJtvIHOGQloGHgRG6RGw8eZRvFKzHQc9CmDQY5rx8aT5-rb0uWS9YS9ccnco/s200/945520_448664931892903_1321992920_n.jpg" width="200" /></a></div>
<b><span style="font-size: 16px;">Eg 1) </span><span style="font-size: 16px;">36</span><sup>2 </sup></b><br />
Step 1: Choose the base number.36 is closer to 40. Therefore 40 is the base number.<br />
Step 2:Find how much the given number is more or less of the base number.36 is -4 of 40.<br />
<span style="font-size: 16px;">Step 3: Square the difference. (-4)² =16.</span><br />
Step 4: Add the difference to the given number. 36-4=32.<br />
Step 5: Multiply the above result with the base number. 32 x 40 =1280.<br />
Step 6: Add the result of step 3 with the result of step 5.<br />
1280 + 16=1296.<br />
<b>Ans: 1296.</b><br />
Let’s do with another one.<br />
<b>Eg 2) 62²</b><br />
Step 1: Base number is 60.<br />
Step 2: 62-60=2.<br />
Step 3: 2²=04 (Write 0 in the ten’s place if the square is a single digit)<br />
Step 4: 62+ 2=64.<br />
Step 5: 64 x 60=3840<br />
Step 6: 3840 + 04=3844<br />
<b>Ans:3844.</b><br />
<b>Eg 3) 79²</b><br />
Step 1: Base number is 80.<br />
Step 2: 79-80 =-1.<br />
Step 3: (-1)²=01<br />
Step 4: 79-1=78.<br />
Step 5: 78 x 80=6240.<br />
Step 6: 6240 + 01=<b>6241.</b><br />
<b>Eg 4) 84²</b><br />
Step 1: 80 is the base<br />
Step 2: 84-80 =4<br />
Step 3: (4)²=16<br />
Step 4:84 + 4=88<br />
Step 5: 88 x 80=7040<br />
Step 6: 7040+16=<b>7056</b><br />
<b>Eg 5) 77²</b><br />
Step 1: 80 is the base number.<br />
Step 2: 77-80=-3<br />
Step 3: (-3)²=09<br />
Step 4: 77-3=74<br />
Step 5: 74 x 80 =5920<br />
Step 6:5920 + 09=<b>5929</b><br />
<b>Eg 6) 49²</b><br />
Step 1: 50 is the base number.<br />
Step 2: 49-50=-1<br />
Step 3: (-1)²=01<br />
Step 4: 49-1=48<br />
Step 5: 48 x 50=2400<br />
Step 6: 2400 + 01=<b>2401</b></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-39971872302989476112013-06-05T06:53:00.001-07:002013-06-05T07:15:42.041-07:00A Mind-Blowing 3-Digit Number Squaring Shortcut<div dir="ltr" style="text-align: left;" trbidi="on">
<h1>
</h1>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2nK60fHXZ5sdcAn9P93KIeRcHgdQBJDXdcCMop9phllY5_F9IgfASr5KWJu7RkQvj3iPR1s0cvu_S5SB4He6RQfa-KuzK6CS-ik3SLHdB2wzd8z0obLPgOAxKMm3xuSu2d2SDvYeYqVyJ/s1600/944405_518012974921591_1197754295_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2nK60fHXZ5sdcAn9P93KIeRcHgdQBJDXdcCMop9phllY5_F9IgfASr5KWJu7RkQvj3iPR1s0cvu_S5SB4He6RQfa-KuzK6CS-ik3SLHdB2wzd8z0obLPgOAxKMm3xuSu2d2SDvYeYqVyJ/s200/944405_518012974921591_1197754295_n.jpg" width="200" /></a></div>
<h2>
Now You Can Square Any Number Whose Tens Digit is 5 Without Using a Calculator</h2>
To begin with, I shall first introduce my canny friend, the <i>Rules</i>.
It will be necessary to examine these rules in order to make your way
about this method simpler. Since the tens digit has already been
determined and defined to be 5, the next factor that would really affect
the result of the square would be the value of the hundreds digit. Now
this is where the rule comes in.<br />
<b>Step 1) </b> Assuming the three-digit number to be squared is written in algebraic form, <br /> <b>100<i>a +10b+c</i></b>, where <b><i>b</i></b> is defined as <b>5</b> and both <b><i>a</i></b> and <b><i>c</i></b> are any number from <b>1 - 9</b> and <b>0 - 9</b> respectively. First, we shall deal with <b><i>a</i></b> as in the following:<br />
<b><i>a</i></b> x <b>(<i>a + 1);</i></b> {this will give us the first one or two digits of the answer}<br />
<b>Step 2) </b> Next we shall obtain the following third digit and so on by performing the following: <br /> <b>25 + <i>(2a + 1)</i></b> x <b><i>c;</i></b> {this will give us the following digits of the answer. You may also assume that (<b><i>2a +1</i></b>) is the same as adding the number that comes right after a to <b><i>a</i></b> and multiplying that sum by the last the digit of the number to be squared}<br />
<b>Step 3) </b> Finally, we determine the last two digit of the answer simply by obtaining the square of <b><i>c</i></b> and read the results obtained in step1 and 2 followed by the one obtained in step 3 and that is the complete answer.<br />
<b><i><br /> c<sup>2</sup>;</i></b>
{gives us the last two digits of the answer, hence, it is better to
think that the last two digits are as easy in obtaining as finding the
square of the last digit of the number to be squared}<br />
<b><br /></b><b>Example:</b><br />
a) Find the square of 253.<br />
<b>1)</b> 2 x (2+1) = <b>6</b><br />
<b>2)</b> 25 + (2 x 2 +1) x 3 =<b>40</b><br />
<b>3)</b> 3<sup>2</sup> = <b>09</b><br />
<b>4)</b> Read <b>64009</b> (the answer!)<br />
b) What is the square of 457?<br />
<b>1)</b> 4 x 5 = <b>20 </b> <b>2)</b> 25 + (4 x 2 +1) x 7 = <b>88</b><br />
<b>3)</b> 7<sup>2</sup> = <b>49</b><br />
<b>4)</b> Read <b>208849</b> (the answer!)<br />
c) What is 959<sup>2</sup>?<br />
<b>1)</b> 9 x 10 = <b>90</b><br />
<b>2)</b> 25 + (9 + 10) x 9 = <b>196</b>
{note that the result is a three digit number, hence the value in the
hundreds digit is a carry digit that is added to the result in step 1}<br />
<b>3)</b> 9<sup>2</sup> = <b>81 </b> <b>4)</b> Read <b>919681</b> (the answer!)<br />
<b>Additional Tip:</b> A shortcut to multiplying any single digit number by 19 would be as<br />
simple as the following example showing 19 x 7:<br />
<b>Step1)</b> Take the double of 7 and subtract it by one<br />
That gives us, <b><i>14 - 1 = 13</i></b><br />
<b>Step 2)</b> Subtract 7 from 10. <br /> Which is, <b><i>10 - 7 = 3</i></b><br />
<b>Step 3)</b> Read the result in step 1 followed by the result in step 2 and you will get <b><i>133,</i></b> <br /> which is the answer.<br />
Hence the general rule for <b><i>a</i></b> x <b><i>19</i></b>, would be:<br />
<b>(2a -1) x 10 + (10 - a)</b> {this simply eliminates the hassle of having to deal with carries}<br />
Or alternatively, you may also try; multiplying the number to be
multiplied by 19, by 20 instead and subtract that number again from the
product, <br /> i.e: 7 x 20 - 7 = 133<br />
<h2>
</h2>
</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-14972497093791478802013-06-05T06:19:00.000-07:002013-06-05T06:21:23.117-07:00Multiply Up to 20X20 In Your Head<div dir="ltr" style="text-align: left;" trbidi="on">
<b><span style="color: black; font-family: Times Roman; font-size: small;">
</span></b><br />
<h2>
</h2>
<b>In
just FIVE minutes you should learn to quickly multiply up to 20x20 in your
head.</b> With this trick, you will be able to multiply any two
numbers from 11 to 19 in your head quickly, without the use of a calculator.
<br />
I will assume that you know your multiplication table reasonably well up
to 10x10.
<br />
Try this:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEHr6h7Co0rsNoSopESBG3ZKtVl18Adk24uk8CtRIU-Ek-rfaDZhrkJ9dAHh8B7hrnVv-0PQTIrNjcGCPJMPUraPnFMf_YFC1fIV_wsbjzgpRiHdGEhvuJFLoXODZoCmnozFjp2ZVRY6dH/s1600/13976_483032418419647_1521764671_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEHr6h7Co0rsNoSopESBG3ZKtVl18Adk24uk8CtRIU-Ek-rfaDZhrkJ9dAHh8B7hrnVv-0PQTIrNjcGCPJMPUraPnFMf_YFC1fIV_wsbjzgpRiHdGEhvuJFLoXODZoCmnozFjp2ZVRY6dH/s200/13976_483032418419647_1521764671_n.jpg" width="200" /></a></div>
<br />
<ul>
<li>Take 15 x 13 for an example.
</li>
<li>Always place the larger number of the two on top in your mind.
</li>
<li>First add 15 + 3 = 18
</li>
<li>Add a zero behind it (multiply by 10) to get 180.
</li>
<li>Multiply the covered lower 3 x the single digit above it the "5" (3x5=
15)
</li>
<li>Add 180 + 15 = 195.
</li>
</ul>
<b>That is It! Wasn't that easy? Practice it on paper
first! </b><br />
<br />
<b>Take another example: </b><br />
<br />
<span style="font-size: large;">15 x 16</span><br />
Step1: 15+6 =21<br />
Step2: <span style="font-size: small;">21x10=210</span><br />
<span style="font-size: small;">Step3: 5</span><span style="font-size: small;"><span style="font-size: small;">x6=30</span></span><br />
<span style="font-size: small;"><span style="font-size: small;">Step4: 210+30=240 (answer) </span> </span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-73819631671751160242013-06-05T05:54:00.002-07:002013-06-05T06:25:20.133-07:00Divisibility by prime numbers under 50<div dir="ltr" style="text-align: left;" trbidi="on">
<h1 style="text-align: justify;">
</h1>
<h2 style="text-align: justify;">
</h2>
<div style="text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7ik6JjfUsuVJ4Wo5D9_Pd6VfvXmqiawzRlAj9rm3a6fSJ0sq59zxwnhLQTqM9pN2QtBLLstRqaVNBrPBRgw1eMyOpr1IYHa76NPlD0MNDswvzPCwFb47GQeGSyra-sb74qjPv7Qvc4ze0/s1600/487025_427615180628038_704720622_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7ik6JjfUsuVJ4Wo5D9_Pd6VfvXmqiawzRlAj9rm3a6fSJ0sq59zxwnhLQTqM9pN2QtBLLstRqaVNBrPBRgw1eMyOpr1IYHa76NPlD0MNDswvzPCwFb47GQeGSyra-sb74qjPv7Qvc4ze0/s200/487025_427615180628038_704720622_n.jpg" width="200" /></a></div>
A number is <b>divisible by 2</b> if its last digit is
also (i.e. 0,2,4,6 or 8).
</div>
<div style="text-align: justify;">
A number is <b>divisible by 3</b> if the sum of its digits is also.
Example: 534: 5+3+4=12 and 1+2=3 so 534 is divisible by 3.
</div>
<div style="text-align: justify;">
A number is <b>divisible by 5</b> if the last digit is 5 or 0.
</div>
<div style="text-align: justify;">
<b>Most people know (only) those 3 rules.
Here are my rules for divisibility by the PRIMES up to 50.</b> Why only primes and not also
composite numbers?
A number is divisible by a composite if it is also divisible by all the prime
factors (e.g. is divisible by 21 if divisible by 3 AND by 7).
Small numbers are used in these worked examples, so you could have used a
pocket calculator. But my rules apply to any number of digits, whereas <b>you cannot
test a 30 or more digit number on your pocket calculator otherwise</b>.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 7</b>. Double the last digit and subtract it
from the remaining leading truncated number. If the result is
divisible by 7, then so was the original number. Apply this rule over
and over again as necessary. Example: 826. Twice 6 is 12. So take 12
from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is
divisible by 7 also.
</div>
<div style="text-align: justify;">
There are similar rules for the remaining primes under 50, i.e. 11,13,
17,19,23,29,31,37,41,43 and 47.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 11</b>. Subtract the last digit from the
remaining leading truncated number. If the result is divisible by 11,
then so was the first number. Apply this rule over and over again as
necessary. <br />
Example: 19151--> 1915-1 =1914 -->191-4=187 -->18-7=11, so
yes, 19151 is divisible by 11.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 13</b>. Add four times the last digit to the
remaining leading truncated number. If the result is divisible by 13,
then so was the first number. Apply this rule over and over again as
necessary. <br />
Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78
is 6*13, so 50661 is divisible by 13.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 17</b>. Subtract five times the last digit from the
remaining leading truncated number. If the result is divisible by 17,
then so was the first number. Apply this rule over and over again as
necessary. <br />
Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is
divisible by 17.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 19</b>. Add two times the last digit to the
remaining leading truncated number. If the result is divisible by 19,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
EG: 101156-->10115+2*6=10127-->1012+2*7=1026-->102+2*6=114
and 114=6*19, so 101156 is divisible by 19.
</div>
<div style="text-align: justify;">
My original divisibilty webpage stopped here. However, I have had a number of mails asking for
divisibility tests for larger primes, so I've extended the list up to 50. Actually even
with 37 most people cannot do the necessary mental arithmetic easily, because they cannot
recognise even single-digit multiples of two-digit numbers on sight. People are no longer
taught the multiplication table up to 20*20 as I was as a child. Nowadays we are lucky if they
know it up to 10*10.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 23</b>. 3*23=69, ends in a 9, so ADD. Add 7 times the last digit to the
remaining leading truncated number. If the result is divisible by 23,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 17043-->1704+7*3=1725-->172+7*5=207 which is 9*23,
so 17043 is also divisible by 23.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 29</b>. Add three times the last digit to the
remaining leading truncated number. If the result is divisible by 29,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 15689-->1568+3*9=1595-->159+3*5=174-->17+3*4=29,
so 15689 is also divisible by 29.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 31</b>. Subtract three times the last digit from the
remaining leading truncated number. If the result is divisible by 31,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 7998-->799-3*8=775-->77-3*5=62 which is twice 31,
so 7998 is also divisible by 31.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 37</b>. This is (slightly) more difficult, since it
perforce uses a double-digit multiplier, namely eleven. People can usually do
single digit multiples of 11, so we can use the same technique still.
Subtract eleven times the last digit from the
remaining leading truncated number. If the result is divisible by 37,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 23384-->2338-11*4=2294-->229-11*4=185 which is five times 37,
so 23384 is also divisible by 37.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 41</b>. Subtract four times the last digit from the
remaining leading truncated number. If the result is divisible by 41,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 30873-->3087-4*3=3075-->307-4*5=287-->28-4*7=0, remainder is zero and
so 30873 is also divisible by 41.
</div>
<div style="text-align: justify;">
<b>Test for divisibility by 43</b>. Now it starts to get really
difficult for most people, because
the multiplier to be used is 13, and most people cannot recognise even single digit multiples
of 13 at sight. You may want to make a little list of 13*N first.
Nevertheless, for the sake of completeness, we will use the same method.
Add thirteen times the last digit to the
remaining leading truncated number. If the result is divisible by 43,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 3182-->318+13*2=344-->34+13*4=86 which is recognisably twice 43, and so
3182 is also divisible by 43.
<br />
<b>Update :</b> Bill Malloy has pointed out that, since we are working to <i>modulo</i>43,
instead of adding factor 13 times the last digit, we can subtract 30 times it, because 13+30=43.
Why didn't I think of that!!! :-(
</div>
<div style="text-align: justify;">
Finally, the <b>Test for divisibility by 47</b>. This too is difficult
for most people, because
the multiplier to be used is 14, and most people cannot recognise even
single digit multiples
of 14 at sight. You may want to make a little list of 14*N first.
Nevertheless, for the sake of completeness, we will use the same method.
Subtract fourteen times the last digit from the
remaining leading truncated number. If the result is divisible by 47,
then so was the first number. Apply this rule over and over again as
necessary.
<br />
Example: 34827-->3482-14*7=3384-->338-14*4=282-->28-14*2=0 , remainder is zero and
so 34827 is divisible by 47.
</div>
<div style="text-align: justify;">
I've stopped here at the last prime below 50, for arbitrary but pragmatic reasons
as explained above.
</div>
<div style="text-align: justify;">
Other blogreaders (sadly even people from <i>.edu</i> domains, who should be able to do
the elementary algebra themselves) have asked why I sometimes say ADD and for other primes say SUBTRACT,
and ask where the <i>apparently arbitrary</i> factors come from.
So let us do some algebra to show the method in my madness.
</div>
<div style="text-align: justify;">
We have displayed the recursive divisibility test of number N as f-M*r where f are the
front digits of N, r is the rear digit of N and M is some multiplier.
And we want to see if N is divisible by some prime P.
We need a method to work out the values of M. What you do is to calculate (mentally) the
smallest multiple of P which ends in a 9 or a 1.
If it's a 9 we are going to ADD, Then we will use the leading digit(s) of the multiple +1 as our multiplier M.
If it's a 1 we are going to SUBTRACT later.
then we will use the leading digit(s) of the multiple as our multiplier M.
</div>
<div style="text-align: justify;">
Example for P=17 : three times 17 is 51 which is the smallest multiple of 17 that
ends in a 1 or 9. Since it's a 1 we are going to SUBTRACT later. The leading digit is a 5,
so we are going to SUBTRACT five times the remainder r. The algorithm was stated above.
Now let's do the algebraic proof.
Writing N=10f+r, we can multiply by -5 (as shown in the example for 17), getting
-5N=-50f-5r. Now we add 51f to both sides (because 51 was the smallest multiple of P=17 to
end in a 1 or a 9), giving one f (which we want), so
51f-5N=f-5r. Now if N is divisible by P (here P=17), we can substitute to get
51f-5*17*x=f-5r and rearrange the left side as 17*(3f-5x)=f-5r and therefore
f-5r is a multiple of P=17 also. Q.E.D.
</div>
</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-74787588410927532882013-05-30T05:11:00.001-07:002013-05-30T09:49:24.483-07:00Squaring numbers ending in 1 <div dir="ltr" style="text-align: left;" trbidi="on">
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOL6crqDYZ5pFL6wtTLPDFqIYYPOrlyvRhnzRu-3rmU6XMzxFCtSpf2zBPEiL-9ummHW5b6iWE5Prp4Yxib0LFcTb6VA1ngFgDJEc3ItUkFzfpyKq6H-SCZgtzBMtmX6Zc6Vn0B7cojLW0/s1600/13976_483032418419647_1521764671_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOL6crqDYZ5pFL6wtTLPDFqIYYPOrlyvRhnzRu-3rmU6XMzxFCtSpf2zBPEiL-9ummHW5b6iWE5Prp4Yxib0LFcTb6VA1ngFgDJEc3ItUkFzfpyKq6H-SCZgtzBMtmX6Zc6Vn0B7cojLW0/s200/13976_483032418419647_1521764671_n.jpg" width="200" /></a></div>
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Here’s
a simple trick to square any number that ends in 1.</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<ul style="margin-top: 0in; text-align: justify;" type="disc">
<li class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Subtract
1 from the number. </span></span></li>
<li class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Square
the difference. (Squaring of such number is easy as it ends in ‘0’)</span></span></li>
<li class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Add
the difference twice to its square. </span></span></li>
<li class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Add
1.</span></span></li>
</ul>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">
</span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Example: If the number to be squared is<b> 61</b></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">
</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">1.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Subtract 1=> 61 - 1 = 60.</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">2.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Square the difference => 60 × 60 =
3600.</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">3.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Add the difference twice to its square
=> 3600 + 60 + 60 = 3720.</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">4.<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Add 1 => 3720 + 1 = 3721.</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify; text-indent: 0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">So, 61<sup>2</sup>
= 3721</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
<span style="font-size: x-small;"><b><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">81<sup>2</sup>
=?</span></b></span></div>
<a href="http://www.blogger.com/null" name="more"></a><br />
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">1)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">81 - 1 = 80 (Subtract 1). </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">2)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">80<sup>2</sup> = 6400 (square the
difference). </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">3)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">6400 + 80 + 80 = 6560 (add the
difference twice to its square).</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">4)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">6560 + 1 = 6561 (add 1) </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify; text-indent: 0.25in;">
</div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify; text-indent: 0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">So, 81<sup>2</sup>
= 6561</span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;"></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">If
the number to be squared is a 3-digit number, let’s say <b>121</b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">1)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">121 - 1 = 120 (Subtract 1). </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">2)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">120<sup>2</sup> = 14400 (square the
difference). </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">3)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">14400 + 120 + 120 = 14640 (add the
difference twice to its square).</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">4)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">14640 + 1 = 14641 (add 1) </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify;">
</div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify; text-indent: 0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">So, 121<sup>2</sup>
= 14641</span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;"></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
<span style="font-size: x-small;"><b><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">251<sup>2</sup>
=?</span></b></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: justify;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">1)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">251 - 1 = 250 </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">2)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">250<sup>2</sup> = 62500 </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">3)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">62500 + 250 + 250 = 63000</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify; text-indent: -0.25in;">
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">4)<span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-family: "Times New Roman"; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">
</span></span><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">63000 + 1 = 63001 </span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0.0001pt 0.5in; text-align: justify;">
</div>
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">So, 251<sup>2</sup>
= 63001</span></span><br />
<br />
<span style="font-size: x-small;"><span style="font-family: "Verdana","sans-serif"; line-height: 115%;">Ref. <a href="http://faster-maths.blogspot.com/2013/05/squaring-numbers-ending-in-1.html#.UadByNjKWVo">http://faster-maths.blogspot.com/2013/05/squaring-numbers-ending-in-1.html#.UadByNjKWVo</a> </span></span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-12953542329175109362013-05-12T21:40:00.002-07:002013-06-05T06:33:24.823-07:00Divisibility Rules<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="text-align: left;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN__I8WjhyFZsNDxiFNMrDGXowep48nZ4CbpB48kF-AvPnluDq96E91DXcmWR6l9ZPjXi6YKicXOJzmjn36KhOZzXj6Nq09bkK8cou6YfHDoUPKbRQ6-LZAn31bbdXkXKtRMxRp_v4Z9Qp/s1600/399758_506547612734794_927267729_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiN__I8WjhyFZsNDxiFNMrDGXowep48nZ4CbpB48kF-AvPnluDq96E91DXcmWR6l9ZPjXi6YKicXOJzmjn36KhOZzXj6Nq09bkK8cou6YfHDoUPKbRQ6-LZAn31bbdXkXKtRMxRp_v4Z9Qp/s200/399758_506547612734794_927267729_n.jpg" width="200" /></a></div>
<b>Dividing by 3</b>
</div>
<ul style="text-align: left;">Add up the digits: if the sum is divisible by three, then the number is as well.
<b>Examples</b>:
<ol>
<li>111111: the digits add to 6 so the whole number is divisible by three. </li>
<li>87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three.
</li>
</ol>
<b>Why does the 'divisibility by 3' rule work?</b><br />
Look at a 2 digit number: 10a+b=9a+(a+b). We know that 9a is divisible by
3, so 10a+b will be divisible by 3 if and only if a+b is. Similarly,
100a+10b+c=99a+9b+(a+b+c), and 99a+9b is divisible by 3, so the total will
be if a+b+c is. <br />
This explanation also works to prove the divisibility by 9 test.
It is originates from modular arithmetic ideas.
</ul>
<div style="text-align: left;">
<b>Dividing by 4</b>
</div>
<ul style="text-align: left;">Look at the last two digits. If the number formed by its last two digits is divisible by 4, the original number is as well. <br /><b>Examples</b>:
<ol>
<li>100 is divisible by 4.
</li>
<li>1732782989264864826421834612 is divisible by four also, because 12 is divisible by four.
</li>
</ol>
</ul>
<div style="text-align: left;">
<b>Dividing by 5</b>
</div>
<ul style="text-align: left;">If the last digit is a five or a zero, then the number is divisible by 5.
</ul>
<div style="text-align: left;">
<b>Dividing by 6</b>
</div>
<ul style="text-align: left;">Check 3 and 2. If the number is divisible by both 3 and 2, it is divisible by 6 as well.
</ul>
<ul style="text-align: left;">Another easy way to tell if a [multi-digit] number is divisible by six . . .
is to look at its [ones digit]: if it is even, and the sum of the [digits] is
a multiple of 3, then the number is divisible by 6.
</ul>
<div style="text-align: left;">
<b>Dividing by 7</b>
</div>
<ul style="text-align: left;">To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number. <br /><b>Example</b>: If you had 203, you would double the last digit to get six,
and subtract that from 20 to get 14. If you get an answer divisible by 7
(including zero), then the original number is divisible by seven. If you
don't know the new number's divisibility, you can apply the rule again.
</ul>
<ul style="text-align: left;">Another methods is:
<blockquote>
<pre>To know if a number is a multiple of seven or not, we can use also
3 coefficients (1 , 2 , 3). We multiply the first number starting
from the ones place by 1, then the second from the right by 3,
the third by 2, the fourth by -1, the fifth by -3, the sixth by -2,
and the seventh by 1, and so forth.
Example: 348967129356876.
6 + 21 + 16 - 6 - 15 - 6 + 9 + 6 + 2 - 7 - 18 - 18 + 8 + 12 + 6 = 16
means the number is not multiple of seven.
If the number was 348967129356874, then the number is a multiple </pre>
</blockquote>
of seven
because instead of 16, we would find 14 as a result, which is a<br />
multiple of 7.
So the pattern is as follows: for a number onmlkjihgfedcba,<br />
calculate
a + 3b + 2c - d - 3e - 2f + g + 3h + <br />
<br />
<blockquote>
<pre>2i - j - 3k - 2l + m + 3n + 2o.
Example: 348967129356874.
Below each digit let me write its respective figure.
3 4 8 9 6 7 1 2 9 3 5 6 8 7 6
2 3 1 -2 -3 -1 2 3 1 -2 -3 -1 2 3 1
(3×2) + (4×3) + (8×1) + (9×-2) + (6×-3) + (7×-1) +
(1×2) + (2×3) + (9×1) + (3×-2) + (5×-3) + (6×-1) +
(8×2) + (7×3) + (6×1) = 16 -- not a multiple of seven.
</pre>
</blockquote>
</ul>
<div style="text-align: left;">
<b>Dividing by 8</b>
</div>
<ul style="text-align: left;">Check the last three digits. Since 1000 is divisible by 8, if
the last three digits of a number are divisible by 8, then so is the
whole number.<br /><b>Example</b>: <b>33333888</b> is divisible by 8; 33333886 isn't.
How can you tell whether the last three digits are divisible by 8?<br />
If the first digit is even, the number is divisible by 8 if the last two
digits are. If the first digit is odd, subtract 4 from the last two digits;
the number will be divisible by 8 if the resulting last two digits are. So,
to continue the last example,
<b>33333888</b> is divisible by 8 because the digit in the hundreds place is an
even number, and the last two digits are 88, which is divisible by 8.
33333886 is not divisible by 8 because the digit in the hundreds place is an
even number, but the last two digits are 86, which is not divisible by 8.
</ul>
<div style="text-align: left;">
<b>Dividing by 9</b>
</div>
<ul style="text-align: left;">Add the digits. If that sum is divisible by nine, then the original number is as well. This holds for any power of three.</ul>
<div style="text-align: left;">
<b>Dividing by 10</b>
</div>
<ul style="text-align: left;">If the number ends in 0, it is divisible by 10.</ul>
<div style="text-align: left;">
<b>Dividing by 11</b>
</div>
<ul style="text-align: left;">Take any number, such as <b>365167484</b>.
Add the first, third, fifth, seventh,.., </ul>
<ul style="text-align: left;">digits.....3 + 5 + 6 + 4 + 4 = <b>22</b>
<br />
Add the second, fourth, sixth, eighth,.., digits.....6 + 1 + 7 + 8 = <b>22</b>
<br />
If the difference, including 0, is divisible by 11,
then so is the number. <br />
<b>22 - 22 = 0</b> so <b>365167484</b> is evenly divisible by <b>11</b>.<br />
</ul>
<div style="text-align: left;">
<b>Dividing by 12</b>
</div>
<ul style="text-align: left;">
Check for divisibility by 3 and 4.
</ul>
</div>
Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-4677485644861377001.post-40811191236553323792013-05-12T21:38:00.002-07:002013-06-05T06:26:38.764-07:00Number Notations for Maths<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajxO50CvCytEqaGuGXbvNAfE6My2Fl1uV5H77twdPGZiXXje1RSBc8w5zlApfbpdEtNbiMm2IcWT7a5KVF8_khaJPg0lSo18UTz9bwdgqirTpGJ7W0wj9fz0IH3hilC55WohWfEFiiH9H/s1600/969349_516267975096091_608622639_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajxO50CvCytEqaGuGXbvNAfE6My2Fl1uV5H77twdPGZiXXje1RSBc8w5zlApfbpdEtNbiMm2IcWT7a5KVF8_khaJPg0lSo18UTz9bwdgqirTpGJ7W0wj9fz0IH3hilC55WohWfEFiiH9H/s200/969349_516267975096091_608622639_n.jpg" width="149" /></a></div>
<span style="font-family: ARIAL; font-size: medium;">
</span>
<span style="font-family: ARIAL; font-size: medium;"><span style="color: maroon;">Hierarchy of Decimal Numbers</span> </span><br />
<span style="font-family: ARIAL; font-size: medium;">
<table border="1" cellpadding="2" cellspacing="0" style="width: 90%px;">
<tbody>
<tr bgcolor="#99CCFF">
<td><div align="left">
<b>Number</b></div>
</td>
<td><div align="left">
<b>Name</b></div>
</td>
<td><div align="left">
<b>How many</b></div>
</td>
</tr>
<tr>
<td>0 </td>
<td>zero</td>
<td></td>
</tr>
<tr>
<td>1 </td>
<td>one</td>
<td></td>
</tr>
<tr>
<td>2 </td>
<td>two</td>
<td></td>
</tr>
<tr>
<td>3 </td>
<td>three</td>
<td></td>
</tr>
<tr>
<td>4 </td>
<td>four</td>
<td></td>
</tr>
<tr>
<td>5 </td>
<td>five</td>
<td></td>
</tr>
<tr>
<td>6 </td>
<td>six</td>
<td></td>
</tr>
<tr>
<td>7 </td>
<td>seven</td>
<td></td>
</tr>
<tr>
<td>8 </td>
<td>eight</td>
<td></td>
</tr>
<tr>
<td>9 </td>
<td>nine</td>
<td></td>
</tr>
<tr>
<td>10 </td>
<td>ten</td>
<td></td>
</tr>
<tr>
<td>20</td>
<td>twenty</td>
<td>two tens</td>
</tr>
<tr>
<td>30</td>
<td>thirty</td>
<td>three tens</td>
</tr>
<tr>
<td>40</td>
<td>forty</td>
<td>four tens</td>
</tr>
<tr>
<td>50 </td>
<td>fifty</td>
<td>five tens</td>
</tr>
<tr>
<td>60</td>
<td>sixty</td>
<td>six tens</td>
</tr>
<tr>
<td>70</td>
<td>seventy</td>
<td>seven tens</td>
</tr>
<tr>
<td>80</td>
<td>eighty</td>
<td>eight tens</td>
</tr>
<tr>
<td>90</td>
<td>ninety</td>
<td>nine tens</td>
</tr>
</tbody></table>
<br />
<table border="1" cellpadding="2" cellspacing="0" style="width: 90%px;">
<tbody>
<tr bgcolor="#99CCFF">
<td width="23%"><b>Number</b></td>
<td width="44%"><b>Name</b></td>
<td width="33%"><b>How Many</b></td>
</tr>
<tr>
<td width="23%">100 </td>
<td width="44%">one hundred</td>
<td width="33%">ten tens</td>
</tr>
<tr>
<td width="23%">1,000 </td>
<td width="44%">one thousand</td>
<td width="33%">ten hundreds</td>
</tr>
<tr>
<td width="23%">10,000</td>
<td width="44%">ten thousand</td>
<td width="33%">ten thousands</td>
</tr>
<tr>
<td width="23%">100,000</td>
<td width="44%">one hundred thousand</td>
<td width="33%">one hundred thousands</td>
</tr>
<tr>
<td width="23%">1,000,000</td>
<td width="44%">one million</td>
<td width="33%">one thousand thousands</td>
</tr>
</tbody></table>
Some people use a comma to mark every 3 digits. It just keeps track of the digits and makes the numbers easier to read.<br />
Beyond a million, the names of the numbers
differ depending where you live. The places are grouped by thousands in
America and France,
by the millions in Great Britain and Germany.<br />
<br />
<table border="1" cellpadding="2" cellspacing="0" style="width: 90%px;">
<tbody>
<tr bgcolor="#99CCFF">
<td><b>Name </b></td>
<td><b>American-French</b></td>
<td><b>English-German</b></td>
</tr>
<tr>
<td>million</td>
<td>1,000,000</td>
<td>1,000,000</td>
</tr>
<tr>
<td>billion</td>
<td>1,000,000,000 (a thousand millions)</td>
<td>1,000,000,000,000 (a million millions)</td>
</tr>
<tr>
<td>trillion</td>
<td>1 with 12 zeros</td>
<td>1 with 18 zeros</td>
</tr>
<tr>
<td>quadrillion</td>
<td>1 with 15 zeros</td>
<td>1 with 24 zeros</td>
</tr>
<tr>
<td>quintillion</td>
<td>1 with 18 zeros</td>
<td>1 with 30 zeros</td>
</tr>
<tr>
<td>sextillion</td>
<td>1 with 21 zeros</td>
<td>1 with 36 zeros</td>
</tr>
<tr>
<td>septillion</td>
<td>1 with 24 zeros</td>
<td>1 with 42 zeros</td>
</tr>
<tr>
<td>octillion</td>
<td>1 with 27 zeros</td>
<td>1 with 48 zeros</td>
</tr>
<tr>
<td>googol</td>
<td colspan="2"><div align="center">
1 with 100 zeros</div>
</td>
</tr>
<tr>
<td>googolplex</td>
<td colspan="2"><div align="center">
1 with a google of zeros</div>
</td>
</tr>
</tbody></table>
<span style="color: lime;">Fractions<br />
</span>Digits to the right of the decimal point
represent the fractional part of the decimal number. Each place value
has a value that is one tenth the value to the immediate left of it.<br />
<table border="1" cellpadding="2" cellspacing="1" style="width: 90%px;">
<tbody>
<tr bgcolor="#99CCFF">
<td><b>Number</b></td>
<td><b>Name</b></td>
<td><b>Fraction</b></td>
</tr>
<tr>
<td>.1</td>
<td>tenth</td>
<td>1/10</td>
</tr>
<tr>
<td>.01</td>
<td>hundredth</td>
<td>1/100</td>
</tr>
<tr>
<td>.001</td>
<td>thousandth</td>
<td>1/1000</td>
</tr>
<tr>
<td>.0001</td>
<td>ten thousandth</td>
<td>1/10000</td>
</tr>
<tr>
<td>.00001</td>
<td>hundred thousandth</td>
<td>1/100000</td>
</tr>
</tbody></table>
<b>Examples:</b><br />
0.234 = 234/1000 (said - point 2 3 4, or 234 thousandths, or two hundred thirty four thousandths)<br />
4.83 = 4 83/100 (said - 4 point 8 3, or 4 and 83 hundredths)<br />
<span style="color: maroon;"><span style="color: lime;">SI Prefixes</span></span> <br />
<table bgcolor="#00c0ff" border="1" cellpadding="1" cellspacing="0">
<tbody>
<tr>
<td><table bgcolor="#ffffff" style="height: 100px;">
<tbody>
<tr bgcolor="#99CCFF">
<td><b>Number</b></td>
<td><b>Prefix</b></td>
<td><b>Symbol</b></td>
</tr>
<tr>
<td>10 <sup>1</sup></td>
<td>deka-</td>
<td>da</td>
</tr>
<tr>
<td>10 <sup>2</sup></td>
<td>hecto-</td>
<td>h</td>
</tr>
<tr>
<td>10 <sup>3</sup></td>
<td>kilo-</td>
<td>k</td>
</tr>
<tr>
<td>10 <sup>6</sup></td>
<td>mega-</td>
<td>M</td>
</tr>
<tr>
<td>10 <sup>9</sup></td>
<td>giga-</td>
<td>G</td>
</tr>
<tr>
<td>10 <sup>12</sup></td>
<td>tera-</td>
<td>T</td>
</tr>
<tr>
<td>10 <sup>15</sup></td>
<td>peta-</td>
<td>P</td>
</tr>
<tr>
<td>10 <sup>18</sup></td>
<td>exa-</td>
<td>E</td>
</tr>
<tr>
<td>10 <sup>21</sup></td>
<td>zeta-</td>
<td>Z</td>
</tr>
<tr>
<td>10 <sup>24</sup></td>
<td>yotta-</td>
<td>Y</td>
</tr>
</tbody></table>
</td>
<td><table bgcolor="#ffffff">
<tbody>
<tr bgcolor="#99CCFF">
<td><b>Number</b></td>
<td><b>Prefix</b></td>
<td><b>Symbol</b></td>
</tr>
<tr>
<td>10 <sup>-1</sup></td>
<td>deci-</td>
<td>d</td>
</tr>
<tr>
<td>10 <sup>-2</sup></td>
<td>centi-</td>
<td>c</td>
</tr>
<tr>
<td>10 <sup>-3</sup></td>
<td>milli-</td>
<td>m</td>
</tr>
<tr>
<td>10 <sup>-6</sup></td>
<td>micro-</td>
<td>u (greek mu)</td>
</tr>
<tr>
<td>10 <sup>-9</sup></td>
<td>nano-</td>
<td>n</td>
</tr>
<tr>
<td>10 <sup>-12</sup></td>
<td>pico-</td>
<td>p</td>
</tr>
<tr>
<td>10 <sup>-15</sup></td>
<td>femto-</td>
<td>f</td>
</tr>
<tr>
<td>10 <sup>-18</sup></td>
<td>atto-</td>
<td>a</td>
</tr>
<tr>
<td>10 <sup>-21</sup></td>
<td>zepto-</td>
<td>z</td>
</tr>
<tr>
<td>10 <sup>-24</sup></td>
<td>yocto-</td>
<td>y</td>
</tr>
</tbody></table>
</td>
</tr>
</tbody></table>
<br />
<span style="color: maroon;"><span style="color: lime;">Roman Numerals</span></span> <br />
<table border="1" cellpadding="2" cellspacing="0" style="width: 277px;">
<tbody>
<tr>
<td valign="bottom">I=1</td>
<td></td>
<td>(I with a bar is not used)</td>
</tr>
<tr>
<td valign="bottom">V=5</td>
<td></td>
<td>_<br />
V=5,000</td>
</tr>
<tr>
<td valign="bottom">X=10</td>
<td></td>
<td>_<br />
X=10,000</td>
</tr>
<tr>
<td valign="bottom">L=50</td>
<td></td>
<td>_<br />
L=50,000</td>
</tr>
<tr>
<td valign="bottom">C=100</td>
<td></td>
<td>_<br />
C = 100 000</td>
</tr>
<tr>
<td valign="bottom">D=500</td>
<td></td>
<td>_<br />
D=500,000</td>
</tr>
<tr>
<td valign="bottom">M=1,000</td>
<td></td>
<td>_<br />
M=1,000,000</td>
</tr>
</tbody></table>
<br />
<b>Examples:</b> <br />
<table border="1" cellpadding="3" cellspacing="0">
<tbody>
<tr>
<td><pre>1 = I
2 = II
3 = III
4 = IV
5 = V
6 = VI
7 = VII
8 = VIII
9 = IX
10 = X</pre>
</td>
<td><pre>11 = XI
12 = XII
13 = XIII
14 = XIV
15 = XV
16 = XVI
17 = XVII
18 = XVIII
19 = XIX
20 = XX
21 = XXI</pre>
</td>
<td><pre>25 = XXV
30 = XXX
40 = XL
49 = XLIX
50 = L
51 = LI
60 = LX
70 = LXX
80 = LXXX
90 = XC
99 = XCIX
</pre>
</td>
</tr>
</tbody></table>
There is no zero in the roman numeral system.<br />
The numbers are built starting from the largest number
on the left, and adding smaller numbers to the right. All the numerals
are then added together.<br />
The exception is the subtracted numerals, if a numeral
is before a larger numeral, you subtract the first numeral from the second.
That is, IX is 10 - 1= 9.<br />
This only works for one small numeral before
one larger numeral - for example, IIX is not 8, it is not a recognized
roman numeral.<br />
There is no place value in this system - the number III
is 3, not 111.<br />
<span style="color: maroon;"><span style="color: lime;">Number Base Systems</span></span>
<table border="1" cellpadding="2" cellspacing="1" style="width: 90%px;">
<tbody>
<tr bgcolor="#99CCFF">
<td><div align="center">
<b>Decimal(10)</b></div>
</td>
<td><div align="center">
<b>Binary(2)</b></div>
</td>
<td><div align="center">
<b>Ternary(3)</b></div>
</td>
<td><div align="center">
<b>Octal(8)</b></div>
</td>
<td><div align="center">
<b>Hexadecimal(16)</b></div>
</td>
</tr>
<tr>
<td><div align="right">
0</div>
</td>
<td><div align="right">
0</div>
</td>
<td><div align="right">
0</div>
</td>
<td><div align="right">
0</div>
</td>
<td><div align="right">
0</div>
</td>
</tr>
<tr>
<td><div align="right">
1</div>
</td>
<td><div align="right">
1</div>
</td>
<td><div align="right">
1</div>
</td>
<td><div align="right">
1</div>
</td>
<td><div align="right">
1</div>
</td>
</tr>
<tr>
<td><div align="right">
2</div>
</td>
<td><div align="right">
10</div>
</td>
<td><div align="right">
2</div>
</td>
<td><div align="right">
2</div>
</td>
<td><div align="right">
2</div>
</td>
</tr>
<tr>
<td><div align="right">
3</div>
</td>
<td><div align="right">
11</div>
</td>
<td><div align="right">
10</div>
</td>
<td><div align="right">
3</div>
</td>
<td><div align="right">
3</div>
</td>
</tr>
<tr>
<td><div align="right">
4</div>
</td>
<td><div align="right">
100</div>
</td>
<td><div align="right">
11</div>
</td>
<td><div align="right">
4</div>
</td>
<td><div align="right">
4</div>
</td>
</tr>
<tr>
<td><div align="right">
5</div>
</td>
<td><div align="right">
101</div>
</td>
<td><div align="right">
12</div>
</td>
<td><div align="right">
5</div>
</td>
<td><div align="right">
5</div>
</td>
</tr>
<tr>
<td><div align="right">
6</div>
</td>
<td><div align="right">
110</div>
</td>
<td><div align="right">
20</div>
</td>
<td><div align="right">
6</div>
</td>
<td><div align="right">
6</div>
</td>
</tr>
<tr>
<td><div align="right">
7</div>
</td>
<td><div align="right">
111</div>
</td>
<td><div align="right">
21</div>
</td>
<td><div align="right">
7</div>
</td>
<td><div align="right">
7</div>
</td>
</tr>
<tr>
<td><div align="right">
8</div>
</td>
<td><div align="right">
1000</div>
</td>
<td><div align="right">
22</div>
</td>
<td><div align="right">
10</div>
</td>
<td><div align="right">
8</div>
</td>
</tr>
<tr>
<td><div align="right">
9</div>
</td>
<td><div align="right">
1001</div>
</td>
<td><div align="right">
100</div>
</td>
<td><div align="right">
11</div>
</td>
<td><div align="right">
9</div>
</td>
</tr>
<tr>
<td><div align="right">
10</div>
</td>
<td><div align="right">
1010</div>
</td>
<td><div align="right">
101</div>
</td>
<td><div align="right">
12</div>
</td>
<td><div align="right">
A</div>
</td>
</tr>
<tr>
<td><div align="right">
11</div>
</td>
<td><div align="right">
1011</div>
</td>
<td><div align="right">
102</div>
</td>
<td><div align="right">
13</div>
</td>
<td><div align="right">
B</div>
</td>
</tr>
<tr>
<td><div align="right">
12</div>
</td>
<td><div align="right">
1100</div>
</td>
<td><div align="right">
110</div>
</td>
<td><div align="right">
14</div>
</td>
<td><div align="right">
C</div>
</td>
</tr>
<tr>
<td><div align="right">
13</div>
</td>
<td><div align="right">
1101</div>
</td>
<td><div align="right">
111</div>
</td>
<td><div align="right">
15</div>
</td>
<td><div align="right">
D</div>
</td>
</tr>
<tr>
<td><div align="right">
14</div>
</td>
<td><div align="right">
1110</div>
</td>
<td><div align="right">
112</div>
</td>
<td><div align="right">
16</div>
</td>
<td><div align="right">
E</div>
</td>
</tr>
<tr>
<td><div align="right">
15</div>
</td>
<td><div align="right">
1111</div>
</td>
<td><div align="right">
120</div>
</td>
<td><div align="right">
17</div>
</td>
<td><div align="right">
F</div>
</td>
</tr>
<tr>
<td><div align="right">
16</div>
</td>
<td><div align="right">
10000</div>
</td>
<td><div align="right">
121</div>
</td>
<td><div align="right">
20</div>
</td>
<td><div align="right">
10</div>
</td>
</tr>
<tr>
<td><div align="right">
17</div>
</td>
<td><div align="right">
10001</div>
</td>
<td><div align="right">
122</div>
</td>
<td><div align="right">
21</div>
</td>
<td><div align="right">
11</div>
</td>
</tr>
<tr>
<td><div align="right">
18</div>
</td>
<td><div align="right">
10010</div>
</td>
<td><div align="right">
200</div>
</td>
<td><div align="right">
22</div>
</td>
<td><div align="right">
12</div>
</td>
</tr>
<tr>
<td><div align="right">
19</div>
</td>
<td><div align="right">
10011</div>
</td>
<td><div align="right">
201</div>
</td>
<td><div align="right">
23</div>
</td>
<td><div align="right">
13</div>
</td>
</tr>
<tr>
<td><div align="right">
20</div>
</td>
<td><div align="right">
10100</div>
</td>
<td><div align="right">
202</div>
</td>
<td><div align="right">
24</div>
</td>
<td><div align="right">
14</div>
</td>
</tr>
</tbody></table>
Each digit can only count up to the value of
one less than the base. In hexadecimal, the letters A - F are used to
represent the digits 10 - 15, so they would only use one character.<br />
</span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-67144772967860000902013-05-12T21:37:00.001-07:002013-06-05T06:27:26.548-07:00Subtraction Tricks<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="font-family: ARIAL; font-size: medium;">
</span>
<br />
<div align="justify">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaQiyiJ-BWsz8-qpixaTyWltyHPcxiNzoYQy-_g8knZyE0c7Fs04Gzukn3LcP3nFn7g8AAVu60_xt5sXnOw0dRK1FqWI48CdmvMGJsm9j4cr6gtt6cbypW3VqEPAxsgffu2D63YS2PJC6u/s1600/947025_516272068429015_962259063_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaQiyiJ-BWsz8-qpixaTyWltyHPcxiNzoYQy-_g8knZyE0c7Fs04Gzukn3LcP3nFn7g8AAVu60_xt5sXnOw0dRK1FqWI48CdmvMGJsm9j4cr6gtt6cbypW3VqEPAxsgffu2D63YS2PJC6u/s200/947025_516272068429015_962259063_n.jpg" width="186" /></a></div>
<span style="font-family: ARIAL; font-size: medium;"><br /><span style="font-size: medium;">A</span>n operation of finding an addend by a sum and another addend:
17 – 6 = 11. Here 17 is a <i>minuend</i>, 6 – a <i>subtrahend</i>, 11 – the <i>difference</i>.</span></div>
<span style="font-family: ARIAL; font-size: medium;">
<span style="font-size: large;"><b>Math tricks 1:</b></span>
subtracting by oversubtracting
<br />
251-85<br />
=(151+100)-85<br />
=151+100-85<br />
=151+15<br />
=166<br />
In this example,251 is break into 151 and 100<br />
<span style="font-size: large;"><b>Math tricks 2:</b></span><br />
subtracting by adding to each number
<br />
251-85<br />
=(151+100)-85<br />
=151+100-85<br />
=151+15<br />
=166<br />
In this example,251 is break into 151 and 100<br />
<span style="font-size: large;"><b>Math tricks 3:</b></span><br />
subtracting by subtracting to each number
<br />
831-104<br />
=(831-4)-(104-4)<br />
=827-100<br />
=727<br />
In this example,251 is break into 151 and 100<br />
</span><br />
<hr />
<span style="font-family: ARIAL; font-size: medium;">
<span style="font-size: large;"><b>Easy way to check your answer:</b></span>
<span style="font-size: large;"><b><u>Check the answer by working backward</u></b></span>
<br />
Example:<br />
<table>
<tbody>
<tr>
<td align="right">543<br />
- 295<br />
<hr />
= 248</td>
</tr>
</tbody></table>
Add 248 to 295,and see whether you can get 543 back or not.
<br />
<table>
<tbody>
<tr>
<td align="right">248<br />
+ 295<br />
<hr />
= 543</td>
</tr>
</tbody></table>
If the answer(eg.248) add with one of the numbers(eg.295) give you another numbers(eg.543),then your aswer is correct.</span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-5457674460528864242013-05-12T21:36:00.002-07:002013-06-05T06:27:43.031-07:00Addition Tricks<div dir="ltr" style="text-align: left;" trbidi="on">
<span style="font-family: ARIAL; font-size: medium;">
</span>
<br />
<div align="justify">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajxO50CvCytEqaGuGXbvNAfE6My2Fl1uV5H77twdPGZiXXje1RSBc8w5zlApfbpdEtNbiMm2IcWT7a5KVF8_khaJPg0lSo18UTz9bwdgqirTpGJ7W0wj9fz0IH3hilC55WohWfEFiiH9H/s1600/969349_516267975096091_608622639_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajxO50CvCytEqaGuGXbvNAfE6My2Fl1uV5H77twdPGZiXXje1RSBc8w5zlApfbpdEtNbiMm2IcWT7a5KVF8_khaJPg0lSo18UTz9bwdgqirTpGJ7W0wj9fz0IH3hilC55WohWfEFiiH9H/s200/969349_516267975096091_608622639_n.jpg" width="149" /></a></div>
<span style="font-family: ARIAL; font-size: medium;"><span style="font-size: x-large;"><b><u>Addition</u></b></span>
<br />-an operation of finding a sum of some numbers:
21 + 5 = 26. Here 21 and 5 – <i>addends</i>, 26 – the <i>sum</i>. If addends are changed by places, a sum is saved
the same: 21 + 5 = 26 and 5 + 21 = 26.</span></div>
<span style="font-family: ARIAL; font-size: medium;">
<span style="font-size: large;"><b>Math tricks 1:</b></span>
<br />
Adding by breaking apart a number
<br />
Example:<br />
137+85<br />
=(122+15)+85<br />
=122+100<br />
=222
<br />
In example here,break 137 to 122 and 15.
<br />
<span style="font-size: large;"><b>Math tricks 2:</b></span>
<br />
Adding by overadding
<br />
Example:<br />
247+97<br />
=247+(100-3)<br />
=347-3<br />
=344
<br />
In example here,97 was add to 100 first.<br />
</span><br />
<hr />
<span style="font-family: ARIAL; font-size: medium;">
<span style="font-size: large;"><b>Easy way to check your answer:</b></span>
<span style="font-size: large;"><b><u>Estimating the answer by rounding off numbers</u></b></span>
<br />
Example:<br />
<table>
<tbody>
<tr>
<td align="right">249<br />
159<br />
+ 504<br />
<hr />
= 912</td>
</tr>
</tbody></table>
249 rounds off to 250,159 to 160 and 504 to 500,then add up these figures.
<br />
<table>
<tbody>
<tr>
<td align="right">250<br />
160<br />
+ 500<br />
<hr />
= 910</td>
</tr>
</tbody></table>
The amount that has been estimated is 910,so the answer should be
around 910.If your answer is only 800,then you know it is wrong.</span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-37551788690862128372013-05-12T21:35:00.002-07:002013-06-05T06:28:27.767-07:00History of Maths<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
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<span style="font-size: x-large;"><b>
<u>An overview</u>
</b></span>
<br />
<span style="font-family: ARIAL; font-size: small;">
Mathematics starts with counting. It is not reasonable, however, to
suggest that early counting was mathematics. Only when some record of
the counting was kept and, therefore, some representation of numbers
occurred can mathematics be said to have started.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
In Babylonia mathematics developed from 2000 BC. Earlier a place value
notation number system had evolved over a lengthy period with a number
base of 60. It allowed arbitrarily large numbers and fractions to be
represented and so proved to be the foundation of more high powered
mathematical development.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Number problems such as that of the Pythagorean triples (a,b,c) with
a2+b2 = c2 were studied from at least 1700 BC. Systems of linear
equations were studied in the context of solving number problems.
Quadratic equations were also studied and these examples led to a type
of numerical algebra.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Geometric problems relating to similar figures, area and volume were also studied and values obtained for p.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The Babylonian basis of mathematics was inherited by the Greeks and
independent development by the Greeks began from around 450 BC. Zeno of
Elea's paradoxes led to the atomic theory of Democritus. A more precise
formulation of concepts led to the realisation that the rational numbers
did not suffice to measure all lengths. A geometric formulation of
irrational numbers arose. Studies of area led to a form of integration.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The theory of conic sections show a high point in pure mathematical
study by Apollonius. Further mathematical discoveries were driven by the
astronomy, for example the study of trigonometry.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The major Greek progress in mathematics was from 300 BC to 200 AD. After
this time progress continued in Islamic countries. Mathematics
flourished in particular in Iran, Syria and India. This work did not
match the progress made by the Greeks but in addition to the Islamic
progress, it did preserve Greek mathematics. From about the 11th Century
Adelard of Bath, then later Fibonacci, brought this Islamic mathematics
and its knowledge of Greek mathematics back into Europe.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Major progress in mathematics in Europe began again at the beginning of
the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with
the algebraic solution of cubic and quartic equations. Copernicus and
Galileo revolutionised the applications of mathematics to the study of
the universe.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The progress in algebra had a major psychological effect and enthusiasm
for mathematical research, in particular research in algebra, spread
from Italy to Stevin in Belgium and Viète in France.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The 17th Century saw Napier, Briggs and others greatly extend the power
of mathematics as a calculatory science with his discovery of
logarithms. Cavalieri made progress towards the calculus with his
infinitesimal methods and Descartes added the power of algebraic methods
to geometry.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Progress towards the calculus continued with Fermat, who, together with
Pascal, began the mathematical study of probability. However the
calculus was to be the topic of most significance to evolve in the 17th
Century.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Newton, building on the work of many earlier mathematicians such as his
teacher Barrow, developed the calculus into a tool to push forward the
study of nature. His work contained a wealth of new discoveries showing
the interaction between mathematics, physics and astronomy. Newton's
theory of gravitation and his theory of light take us into the 18th
Century.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
However we must also mention Leibniz, whose much more rigorous approach
to the calculus (although still unsatisfactory) was to set the scene for
the mathematical work of the 18th Century rather than that of Newton.
Leibniz's influence on the various members of the Bernoulli family was
important in seeing the calculus grow in power and variety of
application.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The most important mathematician of the 18th Century was Euler who, in
addition to work in a wide range of mathematical areas, was to invent
two new branches, namely the calculus of variations and differential
geometry. Euler was also important in pushing forward with research in
number theory begun so effectively by Fermat.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Toward the end of the 18th Century, Lagrange was to begin a rigorous
theory of functions and of mechanics. The period around the turn of the
century saw Laplace's great work on celestial mechanics as well as major
progress in synthetic geometry by Monge and Carnot.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The 19th Century saw rapid progress. Fourier's work on heat was of
fundamental importance. In geometry Plücker produced fundamental work on
analytic geometry and Steiner in synthetic geometry.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Non-euclidean geometry developed by Lobachevsky and Bolyai led to
characterisation of geometry by Riemann. Gauss, thought by some to be
the greatest mathematician of all time, studied quadratic reciprocity
and integer congruences. His work in differential geometry was to
revolutionise the topic. He also contributed in a major way to astronomy
and magnetism.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The 19th Century saw the work of Galois on equations and his insight
into the path that mathematics would follow in studying fundamental
operations. Galois' introduction of the group concept was to herald in a
new direction for mathematical research which has continued through the
20th Century.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Cauchy, building on the work of Lagrange on functions, began rigorous
analysis and began the study of the theory of functions of a complex
variable. This work would continue through Weierstrass and Riemann.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Algebraic geometry was carried forward by Cayley whose work on matrices
and linear algebra complemented that by Hamilton and Grassmann. The end
of the 19th Century saw Cantor invent set theory almost single handedly
while his analysis of the concept of number added to the major work of
Dedekind and Weierstrass on irrational numbers
</span><br />
<span style="font-family: ARIAL; font-size: small;">
Analysis was driven by the requirements of mathematical physics and
astronomy. Lie's work on differential equations led to the study of
topological groups and differential topology. Maxwell was to
revolutionise the application of analysis to mathematical physics.
Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It
led to ergodic theory.
</span><br />
<span style="font-family: ARIAL; font-size: small;">
The study of integral equations was driven by the study of
electrostatics and potential theory. Fredholm's work led to Hilbert and
the development of functional analysis.
</span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-79869855830193108022013-05-12T21:33:00.001-07:002013-06-05T07:15:49.418-07:00Division Tricks<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: ARIAL; font-size: medium;"></span><br />
<div align="justify">
<span style="font-family: ARIAL; font-size: medium;">An operation of finding one of factors by a product and another factor:
48÷4 = 12. Here 48 is a <i>dividend</i>, 4 – a <i>divisor</i>, 12 – the <i>quotient</i>. At <i>dividing integers</i> a quotient can be
not a whole number. Then this quotient can be present as a <i>fraction</i>. If a quotient is a whole number, then it is called that numbers are <i>divisible</i>,
i.e. one number is divided <i>without remainder</i> by another. Otherwise, we have a division <i>with remainder</i>. For example, 23 isn’t divided by 4 ;
this case can be written as: 23 = 5×4 + 3. Here 3 is a <i>remainder</i>.</span></div>
<span style="font-family: ARIAL; font-size: medium;">
<span style="font-size: large;"><b>Math tricks 1:</b></span>
<br />
<span style="font-family: ARIAL; font-size: medium;">Dividing by grouping
</span><br />
<span style="font-family: ARIAL; font-size: medium;">345÷15</span><br />
<span style="font-family: ARIAL; font-size: medium;">=(300÷15)+(45÷15)</span><br />
<span style="font-family: ARIAL; font-size: medium;">=20+3</span><br />
<span style="font-family: ARIAL; font-size: medium;">=23
</span><br />
<span style="font-family: ARIAL; font-size: medium;">In example here,separate dividend into two groups:(300÷15)+(45÷15),and add up the answer.
</span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-size: large;"><b>Math tricks 2:</b></span>
</span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">Dividing by augmenting
</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">Example:</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">418÷11</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=(440÷11)-(22÷11)</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=40-2</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=38
</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">In example here,418 was augmented to 440,that is 418=440-22</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">
</span></span>
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-size: large;"><b>Math tricks 3:</b></span>
</span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">Dividing by break apart the divisor
</span></span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">325÷25</span></span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=325÷(5×5)</span></span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=325÷5÷5</span></span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=65÷5</span></span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">=13
</span></span></span><br />
<span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;"><span style="font-family: ARIAL; font-size: medium;">In example here,break 25 to 5 and 5(25=5×5).
</span></span></span></span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-82462904332049738792013-05-12T21:25:00.000-07:002013-06-05T06:30:22.676-07:00 IMPRESS YOUR FRIENDS WITH MENTAL MATH TRICKS <div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-size: x-large;"><b><a href="http://www.glad2teach.co.uk/fast_maths_calculation_tricks.htm">Easy Cal Tricks </a></b></span><br />
<br />
Being able to perform arithmetic quickly and mentally can greatly
boost your self-esteem, especially if you don't consider yourself to be
very good at Math. And, getting comfortable with arithmetic might just
motivate you to dive deeper into other things mathematical.<br />
This article presents nine ideas that will hopefully get you to look
at arithmetic as a game, one in which you can see patterns among numbers
and pick then apply the right trick to quickly doing the calculation.<br />
The tricks in this article all involve multiplication.<br />
Don't be discouraged if the tricks seem difficult at first. Learn one
trick at a time. Read the description, explanation, and examples
several times for each technique you're learning. Then make up some of
your own examples and practice the technique.<br />
As you learn and practice the tricks make sure you check your results
by doing multiplication the way you're used to, until the tricks start
to become second nature. Checking your results is critically important:
the last thing you want to do is learn the tricks incorrectly.<br />
<span id="more-25"></span><br />
<b>1. Multiplying by 9, or 99, or 999</b><br />
<blockquote>
Multiplying by 9 is really multiplying by 10-1.<br />
So, 9x9 is just 9x(10-1) which is 9x10-9 which is 90-9 or 81.</blockquote>
<blockquote>
Let's try a harder example: 46x9 = 46x10-46 = 460-46 = 414.</blockquote>
<blockquote>
One more example: 68x9 = 680-68 = 612.</blockquote>
<blockquote>
To multiply by 99, you multiply by 100-1.<br />
So, 46x99 = 46x(100-1) = 4600-46 = 4554.</blockquote>
<blockquote>
Multiplying by 999 is similar to multiplying by 9 and by 99.<br />
38x999 = 38x(1000-1) = 38000-38 = 37962.</blockquote>
<b>2. Multiplying by 11</b><br />
<blockquote>
To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.<br />
Let me illustrate:<br />
To multiply 436 by 11 go from right to left.<br />
First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.<br />
Write down 9 to the left of 6.<br />
Then add 4 to 3 to get 7. Write down 7.<br />
Then, write down the leftmost digit, 4.<br />
So, 436x11 = is 4796.<br />
Let's do another example: 3254x11.<br />
The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.<br />
One more example, this one involving carrying: 4657x11.<br />
Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).<br />
Going from right to left we write down 7.<br />
Then we notice that 5+7=12.<br />
So we write down 2 and carry the 1.<br />
6+5 = 11, plus the 1 we carried = 12.<br />
So, we write down the 2 and carry the 1.<br />
4+6 = 10, plus the 1 we carried = 11.<br />
So, we write down the 1 and carry the 1.<br />
To the leftmost digit, 4, we add the 1 we carried.<br />
So, 4657x11 = 51227 .</blockquote>
<b>3. Multiplying by 5, 25, or 125</b><br />
<blockquote>
Multiplying by 5 is just multiplying by 10 and then
dividing by 2. Note: To multiply by 10 just add a 0 to the end of the
number.<br />
12x5 = (12x10)/2 = 120/2 = 60.<br />
Another example: 64x5 = 640/2 = 320.<br />
And, 4286x5 = 42860/2 = 21430.<br />
To multiply by 25 you multiply by 100 (just add two 0's to the end of
the number) then divide by 4, since 100 = 25x4. Note: to divide by 4
your can just divide by 2 twice, since 2x2 = 4.<br />
64x25 = 6400/4 = 3200/2 = 1600.<br />
58x25 = 5800/4 = 2900/2 = 1450.<br />
To multiply by 125, you multipy by 1000 then divide by 8 since 8x125 =
1000. Notice that 8 = 2x2x2. So, to divide by 1000 add three 0's to the
number and divide by 2 three times.<br />
32x125 = 32000/8 = 16000/4 = 8000/2 = 4000.<br />
48x125 = 48000/8 = 24000/4 = 12000/2 = 6000.</blockquote>
<b>4. Multiplying together two numbers that differ by a small even number</b><br />
<blockquote>
This trick only works if you've memorized or can quickly
calculate the squares of numbers. If you're able to memorize some
squares and use the tricks described later for some kinds of numbers
you'll be able to quickly multiply together many pairs of numbers that
differ by 2, or 4, or 6.<br />
Let's say you want to calculate 12x14.<br />
When two numbers differ by two their product is always the square of the number in between them minus 1.<br />
12x14 = (13x13)-1 = 168.<br />
16x18 = (17x17)-1 = 288.<br />
99x101 = (100x100)-1 = 10000-1 = 9999<br />
If two numbers differ by 4 then their product is the square of the
number in the middle (the average of the two numbers) minus 4.<br />
11x15 = (13x13)-4 = 169-4 = 165.<br />
13x17 = (15x15)-4 = 225-4 = 221.<br />
If the two numbers differ by 6 then their product is the square of their average minus 9.<br />
12x18 = (15x15)-9 = 216.<br />
17x23 = (20x20)-9 = 391.</blockquote>
<b>5. Squaring 2-digit numbers that end in 5</b><br />
<blockquote>
If a number ends in 5 then its square always ends in 25.
To get the rest of the product take the left digit and multiply it by
one more than itself.<br />
35x35 ends in 25. We get the rest of the product by multiplying 3 by
one more than 3. So, 3x4 = 12 and that's the rest of the product. Thus,
35x35 = 1225.<br />
To calculate 65x65, notice that 6x7 = 42 and write down 4225 as the answer.<br />
85x85: Calculate 8x9 = 72 and write down 7225.</blockquote>
<b>6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10</b><br />
<blockquote>
Let's say you want to multiply 42 by 48. You notice that
the first digit is 4 in both cases. You also notice that the other
digits, 2 and 8, sum to 10. You can then use this trick: multiply the
first digit by one more than itself to get the first part of the answer
and multiply the last digits together to get the second (right) part of
the answer.<br />
An illustration is in order:<br />
To calculate 42x48: Multiply 4 by 4+1. So, 4x5 = 20. Write down 20.<br />
Multiply together the last digits: 2x8 = 16. Write down 16.<br />
The product of 42 and 48 is thus 2016.<br />
Notice that for this particular example you could also have noticed
that 42 and 48 differ by 6 and have applied technique number 4.<br />
Another example: 64x66. 6x7 = 42. 4x6 = 24. The product is 4224.<br />
A final example: 86x84. 8x9 = 72. 6x4 = 24. The product is 7224</blockquote>
<b>7. Squaring other 2-digit numbers</b><br />
<blockquote>
Let's say you want to square 58. Square each digit and
write a partial answer. 5x5 = 25. 8x8 = 64. Write down 2564 to start.
Then, multiply the two digits of the number you're squaring together,
5x8=40.<br />
Double this product: 40x2=80, then add a 0 to it, getting 800.<br />
Add 800 to 2564 to get 3364.<br />
This is pretty complicated so let's do more examples.<br />
32x32. The first part of the answer comes from squaring 3 and 2.<br />
3x3=9. 2x2 = 4. Write down 0904. Notice the extra zeros. It's
important that every square in the partial product have two digits.<br />
Multiply the digits, 2 and 3, together and double the whole thing. 2x3x2 = 12.<br />
Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.<br />
56x56. The partial product comes from 5x5 and 6x6. Write down 2536.<br />
5x6x2 = 60. Add a zero to get 600.<br />
56x56 = 2536+600 = 3136.<br />
One more example: 67x67. Write down 3649 as the partial product.<br />
6x7x2 = 42x2 = 84. Add a zero to get 840.<br />
67x67=3649+840 = 4489.</blockquote>
<b>8. Multiplying by doubling and halving</b><br />
<blockquote>
There are cases when you're multiplying two numbers
together and one of the numbers is even. In this case you can divide
that number by two and multiply the other number by 2. You can do this
over and over until you get to multiplication this is easy for you to
do.<br />
Let's say you want to multiply 14 by 16. You can do this:<br />
14x16 = 28x8 = 56x4 = 112x2 = 224.<br />
Another example: 12x15 = 6x30 = 6x3 with a 0 at the end so it's 180.<br />
48x17 = 24x34 = 12x68 = 6x136 = 3x272 = 816. (Being able to calculate
that 3x27 = 81 in your head is very helpful for this problem.)</blockquote>
<b>9. Multiplying by a power of 2</b><br />
<blockquote>
To multiply a number by 2, 4, 8, 16, 32, or some other
power of 2 just keep doubling the product as many times as necessary. If
you want to multiply by 16 then double the number 4 times since 16 =
2x2x2x2.<br />
15x16: 15x2 = 30. 30x2 = 60. 60x2 = 120. 120x2 = 240.<br />
23x8: 23x2 = 46. 46x2 = 92. 92x2 = 184.<br />
54x8: 54x2 = 108. 108x2 = 216. 216x2 = 432.</blockquote>
Practice these tricks and you'll get good at solving many different
kinds of arithmetic problems in your head, or at least quickly on paper.
Half the fun is identifying which trick to use. Sometimes more than one
trick will apply and you'll get to choose which one is easiest for a
particular problem.<br />
Multiplication can be a great sport! Enjoy.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-88651498086478942652013-04-23T00:30:00.005-07:002013-06-05T06:30:53.868-07:00Cool Mental Math Tricks <div dir="ltr" style="text-align: left;" trbidi="on">
<div style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 12px; text-align: justify;">
<b>
</b></div>
<div style="text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajxO50CvCytEqaGuGXbvNAfE6My2Fl1uV5H77twdPGZiXXje1RSBc8w5zlApfbpdEtNbiMm2IcWT7a5KVF8_khaJPg0lSo18UTz9bwdgqirTpGJ7W0wj9fz0IH3hilC55WohWfEFiiH9H/s1600/969349_516267975096091_608622639_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajxO50CvCytEqaGuGXbvNAfE6My2Fl1uV5H77twdPGZiXXje1RSBc8w5zlApfbpdEtNbiMm2IcWT7a5KVF8_khaJPg0lSo18UTz9bwdgqirTpGJ7W0wj9fz0IH3hilC55WohWfEFiiH9H/s200/969349_516267975096091_608622639_n.jpg" width="149" /></a></div>
<br /></div>
<div style="text-align: justify;">
Math can be terrifying for many people. This list will
hopefully improve your general knowledge of mathematical tricks and
your speed when you need to do math in your head.</div>
<div style="text-align: justify;">
So here's 9 super mental math tricks...</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>1. Multiplying by 9, or 99, or 999</b></h2>
<div style="text-align: justify;">
Multiplying by 9 is really multiplying by 10-1. So, 9×9 is just
9x(10-1) which is 9×10-9 which is 90-9 or 81. Let’s try a harder
example: 46×9 = 46×10-46 = 460-46 = 414. One more example: 68×9 =
680-68 = 612. To multiply by 99, you multiply by 100-1. So, 46×99 =
46x(100-1) = 4600-46 = 4554. Multiplying by 999 is similar to
multiplying by 9 and by 99. 38×999 = 38x(1000-1) = 38000-38 = 37962.</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>2. Multiplying by 11</b></h2>
<div style="text-align: justify;">
To multiply a number by 11 you add pairs of numbers next to each
other, except for the numbers on the edges. Let me illustrate: To
multiply 436 by 11 go from right to left. First write down the 6 then
add 6 to its neighbor on the left, 3, to get 9. Write down 9 to the
left of 6. Then add 4 to 3 to get 7. Write down 7. Then, write down
the leftmost digit, 4. So, 436×11 = is 4796. Let’s do another example:
3254×11. The answer comes from these sums and edge numbers:
(3)(3+2)(2+5)(5+4)(4) = 35794. One more example, this one involving
carrying: 4657×11. Write down the sums and edge numbers:
(4)(4+6)(6+5)(5+7)(7). Going from right to left we write down 7. Then
we notice that 5+7=12. So we write down 2 and carry the 1. 6+5 = 11,
plus the 1 we carried = 12. So, we write down the 2 and carry the 1.
4+6 = 10, plus the 1 we carried = 11. So, we write down the 1 and carry
the 1. To the leftmost digit, 4, we add the 1 we carried. So, 4657×11
= 51227 .</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>3. Multiplying by 5, 25, or 125</b></h2>
<div style="text-align: justify;">
Multiplying by 5 is just multiplying by 10 and then dividing by 2.
Note: To multiply by 10 just add a 0 to the end of the number. 12×5 =
(12×10)/2 = 120/2 = 60. Another example: 64×5 = 640/2 = 320. And,
4286×5 = 42860/2 = 21430. To multiply by 25 you multiply by 100 (just
add two 0’s to the end of the number) then divide by 4, since 100 =
25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 =
4. 64×25 = 6400/4 = 3200/2 = 1600. 58×25 = 5800/4 = 2900/2 = 1450.
To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 =
1000. Notice that 8 = 2×2x2. So, to divide by 1000 add three 0’s to the
number and divide by 2 three times. 32×125 = 32000/8 = 16000/4 =
8000/2 = 4000. 48×125 = 48000/8 = 24000/4 = 12000/2 = 6000.</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b><a name='more'></a>4. Multiplying together two numbers that differ by a small even number</b></h2>
<div style="text-align: justify;">
This trick only works if you’ve memorized or can quickly calculate the
squares of numbers. If you’re able to memorize some squares and use
the tricks described later for some kinds of numbers you’ll be able to
quickly multiply together many pairs of numbers that differ by 2, or 4,
or 6. Let’s say you want to calculate 12×14. When two numbers differ
by two their product is always the square of the number in between them
minus 1. 12×14 = (13×13)-1 = 168. 16×18 = (17×17)-1 = 288. 99×101 =
(100×100)-1 = 10000-1 = 9999 If two numbers differ by 4 then their
product is the square of the number in the middle (the average of the
two numbers) minus 4. 11×15 = (13×13)-4 = 169-4 = 165. 13×17 =
(15×15)-4 = 225-4 = 221. If the two numbers differ by 6 then their
product is the square of their average minus 9. 12×18 = (15×15)-9 =
216. 17×23 = (20×20)-9 = 391.</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>5. Squaring 2-digit numbers that end in 5</b></h2>
<div style="text-align: justify;">
If a number ends in 5 then its square always ends in 25. To get the
rest of the product take the left digit and multiply it by one more
than itself. 35×35 ends in 25. We get the rest of the product by
multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of
the product. Thus, 35×35 = 1225. To calculate 65×65, notice that 6×7 =
42 and write down 4225 as the answer. 85×85: Calculate 8×9 = 72 and
write down 7225.</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10</b></h2>
<div style="text-align: justify;">
Let’s say you want to multiply 42 by 48. You notice that the first
digit is 4 in both cases. You also notice that the other digits, 2 and
8, sum to 10. You can then use this trick: multiply the first digit by
one more than itself to get the first part of the answer and multiply
the last digits together to get the second (right) part of the answer.
An illustration is in order: To calculate 42×48: Multiply 4 by 4+1.
So, 4×5 = 20. Write down 20. Multiply together the last digits: 2×8 =
16. Write down 16. The product of 42 and 48 is thus 2016. Notice that
for this particular example you could also have noticed that 42 and 48
differ by 6 and have applied technique number 4. Another example:
64×66. 6×7 = 42. 4×6 = 24. The product is 4224. A final example: 86×84.
8×9 = 72. 6×4 = 24. The product is 7224</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>7. Squaring other 2-digit numbers</b></h2>
<div style="text-align: justify;">
Let’s say you want to square 58. Square each digit and write a partial
answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply
the two digits of the number you’re squaring together, 5×8=40. Double
this product: 40×2=80, then add a 0 to it, getting 800. Add 800 to 2564
to get 3364. This is pretty complicated so let’s do more examples.
32×32. The first part of the answer comes from squaring 3 and 2. 3×3=9.
2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that
every square in the partial product have two digits. Multiply the
digits, 2 and 3, together and double the whole thing. 2×3x2 = 12. Add a
zero to get 120. Add 120 to the partial product, 0904, and we get 1024.
56×56. The partial product comes from 5×5 and 6×6. Write down 2536.
5×6x2 = 60. Add a zero to get 600. 56×56 = 2536+600 = 3136. One more
example: 67×67. Write down 3649 as the partial product. 6×7x2 = 42×2 =
84. Add a zero to get 840. 67×67=3649+840 = 4489.</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>8. Multiplying by doubling and halving</b></h2>
<div style="text-align: justify;">
There are cases when you’re multiplying two numbers together and one
of the numbers is even. In this case you can divide that number by two
and multiply the other number by 2. You can do this over and over until
you get to multiplication this is easy for you to do. Let’s say you
want to multiply 14 by 16. You can do this: 14×16 = 28×8 = 56×4 = 112×2
= 224. Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s
180. 48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to
calculate that 3×27 = 81 in your head is very helpful for this
problem.)</div>
<div style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b>9. Multiplying by a power of 2</b></h2>
<div style="text-align: justify;">
To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just
keep doubling the product as many times as necessary. If you want to
multiply by 16 then double the number 4 times since 16 = 2×2x2×2.
15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240. 23×8: 23×2 = 46.
46×2 = 92. 92×2 = 184. 54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432. </div>
</div>
Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-4677485644861377001.post-81714869104455157962013-04-23T00:28:00.003-07:002013-06-05T06:31:55.242-07:00Awsome Mental Math tricks<div dir="ltr" style="text-align: left;" trbidi="on">
<h1 style="text-align: justify;">
</h1>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLcn6hBqgOe6e_Ozmxvz6PR4QB5wIeE7o2WPh1PSV8YOf8qqVnweI3x2kOOCnKTYYJ593hCoCdZFXLwzbENJtvIHOGQloGHgRG6RGw8eZRvFKzHQc9CmDQY5rx8aT5-rb0uWS9YS9ccnco/s1600/945520_448664931892903_1321992920_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="168" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLcn6hBqgOe6e_Ozmxvz6PR4QB5wIeE7o2WPh1PSV8YOf8qqVnweI3x2kOOCnKTYYJ593hCoCdZFXLwzbENJtvIHOGQloGHgRG6RGw8eZRvFKzHQc9CmDQY5rx8aT5-rb0uWS9YS9ccnco/s200/945520_448664931892903_1321992920_n.jpg" width="200" /></a></div>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Squaring a two-digit number ending in 5</span></h3>
<div style="text-align: justify;">
<span style="font-size: small;">To find the square of a two-digit number ending in 5, take the first digit, multiply
by <i>itself plus one</i>, and then put "25" after it.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Example:</b> To find 65<sup>2</sup>, compute 6 * 7 = 42, then write "4225" as the answer.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Example:</b> To find 25<sup>2</sup>, compute 2 * 3 = 6, then write "625" as the answer.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Note:</b> This <i>only works</i> if the second digit is 5!
</span></div>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Multiplying two numbers when a round number is halfway between them</span></h3>
<div style="text-align: justify;">
<span style="font-size: small;">Suppose you are multiplying two moderately large numbers, 84 and 76. Note that the number
80 is halfway between the two.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">This means that 84*76 can be rewritten as</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">(80 + 4)(80 - 4)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">which simplifies to 80<sup>2</sup> - 4<sup>2</sup>,
which is easier to figure out: 6400 - 16 = 6384.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Note:</b> This only works if the number halfway between your two original numbers, is a nice round number.
Otherwise, squaring the number is almost as much mental work as multiplying the original numbers!
</span></div>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Multiplying large numbers very close to a round number</span></h3>
<a name='more'></a><span style="font-size: small;">To multiply 1003 * 1004, for example, rewrite the multiplication mentally as:</span><br />
<div style="text-align: justify;">
<span style="font-size: small;">(1000 + 3)(1000 + 4)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">which expands to:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">1000<sup>2</sup> + (3 + 4)*1000 + 3*4</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">which you can mentally work out to 1,007,012. In general,</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">(1000 + a)(1000 + b) = 1,000,000 + (a + b)1000 + a*b</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">and hopefully you'll be able to recognize a similar strategy when multiplying other large numbers that
are "almost round".
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">Multiplying numbers that are just slightly <i>less</i> than a round number works similarly:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">99*98</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= (100 - 1)(100 - 2)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 10000 - 100 - 200 + 2</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 9702
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">As well as, of course, multiplying a number slightly less than a round number, by a number slightly
greater than a round number:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">1001*998</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= (1000 + 1)(1000 - 2)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 1000000 + 1000 - 2000 - 2</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 998998
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Remember</b> to use the right sign for the last term, the "small" one! If the two large numbers you
are multiplying are <i>both slightly greater</i> than a round number, or <i>both slightly less</i>, then
the sign on the final small term will be positive; if <i>one</i> of the two large numbers is slightly
less than a round number and the <i>other</i> one is slightly greater, than the final small term will
be negative.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">Also, in all of the examples above, the two numbers being multiplied were close to the <i>same</i> round
number. Be prepared for cases where the round numbers might be different, e.g.:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">2001*999</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= (2000 + 1)(1000 - 1)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 2000000 + 1000 - 2000 - 1</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 1998999
</span></div>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Powers of 11</span></h3>
<div style="text-align: justify;">
<span style="font-size: small;">Up to the 4th power, the digits in the powers of 11 follow the same pattern as the binomial coefficients:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">11<sup>2</sup> = 121</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">11<sup>3</sup> = 1331</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">11<sup>4</sup> = 14641</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">i.e. if you expand (x + y)<sup>4</sup> for example, the coefficients:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>1</b>x<sup>4</sup> + <b>4</b>x<sup>3</sup>y + <b>6</b>x<sup>2</sup>y<sup>2</sup> + <b>4</b>xy<sup>3</sup> + <b>1</b>y<sup>4</sup></span></div>
<div style="text-align: justify;">
<span style="font-size: small;">exactly match the digits of 11<sup>4</sup> = 14641.</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><br /></span></div>
<div style="text-align: justify;">
<span style="font-size: small;">This makes sense if you consider 11<sup>4</sup> to be an expansion of (10 + 1)<sup>4</sup>:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">(10 + 1)<sup>4</sup> = <b>1</b>*10<sup>4</sup> + <b>4</b>*10<sup>3</sup> + <b>6</b>*10<sup>2</sup> + <b>4</b>*10<sup>1</sup> + <b>1</b></span>
</div>
<h3 style="text-align: justify;">
<span style="font-size: small;">
Geometric series</span></h3>
<div style="text-align: justify;">
<span style="font-size: small;">These are easy to work out if you remember the formula, and <i>because</i> they're easy to work out,
they're likely to show up in mental math:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><br /></span></div>
<table style="margin-left: 0px; margin-right: 0px; text-align: left;">
<tbody>
<tr>
<td><span style="font-size: small;">a + ar + ar<sup>2</sup> + ar<sup>3</sup> + ... =
</span></td>
<td><table cellpadding="0"><tbody>
<tr><td><center>
<span style="font-size: small;">
a</span></center>
</td></tr>
<tr><td><hr />
</td></tr>
<tr><td><span style="font-size: small;">(1 - r)</span></td></tr>
</tbody></table>
</td>
</tr>
</tbody></table>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Example:</b></span></div>
<div style="text-align: justify;">
<span style="font-size: small;">10 + 10/3 + 10/9 + 10/27 + ...</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 10/(1 - 1/3) (by the formula)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 10 / (2/3)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 10*(3/2)</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">= 15
</span></div>
<table border="0" style="margin-left: 0px; margin-right: 0px; text-align: left;"><tbody>
<tr><td></td></tr>
</tbody></table>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Sum of first n integers</span></h3>
<div style="text-align: justify;">
<span style="font-size: small;">Another easy formula likely to show up:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><br /></span></div>
<table style="margin-left: 0px; margin-right: 0px; text-align: left;">
<tbody>
<tr>
<td><span style="font-size: small;">1 + 2 + 3 + ... + n =
</span></td>
<td><table cellpadding="0"><tbody>
<tr><td><center>
<span style="font-size: small;">
n(n+1)</span></center>
</td></tr>
<tr><td><hr />
</td></tr>
<tr><td><center>
<span style="font-size: small;">
2</span></center>
</td></tr>
</tbody></table>
</td>
</tr>
</tbody></table>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Example:</b> 1 + 2 + 3 + ... + 12 = 12*13/2 = 6*13 = 78
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">(Note that to compute 12*13/2, rather than multiplying 12*13 and then dividing by 2, we divided
12 by 2 right away to get 6*13. This can make all the difference in mental math!)
</span></div>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Sum of the first n odd integers</span></h3>
<div style="text-align: justify;">
<span style="font-size: small;">The sum of the first n odd integers is n<sup>2</sup>.
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Example:</b> The sum of the first 4 odd integers: 1 + 3 + 5 + 7 = 16 = 4<sup>2</sup></span>
</div>
<h3 style="text-align: justify;">
<span style="color: red; font-size: small;">Sum of the first n square numbers</span></h3>
<table style="margin-left: 0px; margin-right: 0px; text-align: left;">
<tbody>
<tr>
<td><span style="font-size: small;">1 + 2<sup>2</sup> + 3<sup>2</sup> + ... + n<sup>2</sup> =
</span></td>
<td><table cellpadding="0"><tbody>
<tr><td><center>
<span style="font-size: small;">
n(n+1)(2n+1)</span></center>
</td></tr>
<tr><td><hr />
</td></tr>
<tr><td><center>
<span style="font-size: small;">
6</span></center>
</td></tr>
</tbody></table>
</td>
</tr>
</tbody></table>
<div style="text-align: justify;">
<span style="font-size: small;"><b>Example:</b> The sum of the first 10 square numbers:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">The long way: 1 + 4 + 9 + 6 + 25 + 36 + 49 + 64 + 81 + 100 = 385</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">The faster way: 10*(10+1)(20+1)/6 = 10*11*21/6 = 385
</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">To work out that last expression, by the way:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">10*11*21/6</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">the shortcut is: First look at the numerator, the expression 10*11*21, and look for a number
that can be divided by 2, and divide it by 2. In this case, reduce the 10 to a 5:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">5*11*21</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">Then look for a number that can be divided by 3 -- in this case, 21 -- and divide it by 3:</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">5*11*7</span></div>
<div style="text-align: justify;">
<span style="font-size: small;">The net effect of this (dividing by 2 and then dividing by 3) is, of course, dividing by 6. So
now you only have to work out 5*11*7 = 385, which is easier (although still not easy) to do in your head.
</span></div>
</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4677485644861377001.post-55423041623134071582013-04-15T20:58:00.001-07:002013-06-05T06:36:04.102-07:00Important Notice!<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-size: large;">প্রিয় ছাত্র-ছাত্রীরা, তোমাদের জন্য সুসংবাদ। বাংলাদেশ মেন্টাল ম্যাথ এ্যান্ড আই কিউ ফাউন্ডেশন 2013 সাল থেকে জাতীয় প্রতিযোগিতার আয়োজন করতে যাচ্ছে। এই ব্লগে প্রস্তুতি নেয়ার সবকিছুই থাকবে। পাশাপাশি ভাল প্রস্তুতির জন্য আজই একটি বই সংগ্রহ করে পড়:</span><br />
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