Mental calculus tips..

Moo · #1 $Old$ April 18th, 2008, 01:34 PM

Hello,

1.*1.

This has nothing to do with extraordinary technique, but you gotta think about it (I'm confident that some of you know about it)..

$(10+a)(10+b)=10(10+a)+b(10+a)=10(10+a)+10b+ab=10(10+a+b)+ab$

So multiplying two such numbers is equivalent to :

- sum one of the numbers with the unit digit of the other one
- multiply it by 10
- add the product of the two unit digits.

$19 \times 17= ?$

- $17+9=26$

- $26 \times 10=260$

- $7 \times 9=63$

- $19 \times 17=260+63=323$

I hope this would help at least one person in the future

Enjoy calculations !

flyingsquirrel · #2 $Old$ April 18th, 2008, 01:58 PM

Hello

I didn't know about this hint. Usually, I simply develop $(10+a)(10+b)=100+10(a+b)+ab$ .

In the case of $19 \times 17$ , where 9 and 7 are close to 10, I'd rather do $19\times17=(20-1)(20-3)=400-20(1+3)+3$ which avoids manipulating too large numbers. (yes, I consider 63 as large

)

Moo · #3 $Old$ April 18th, 2008, 02:07 PM

Well, this goes too for 12x15 for example, which involves smaller numbers

I only gave an example with random numbers... Sometimes, there are easier ways of calculating

When I'm talking about mental calculations, I consider it as being without any paper & pen... Unfortunately, (20-1)(20-3) for some people needs some writing :-)

Another thing :

The square of a number whose unit digit is 5.

If you're looking for the square of a number N=X5 (with X representating all the digits of the number, meaning that actually, $N=10X+5$ ), don't worry, this can be shortened ^^

First of all, calculate $X(X+1)=Y$

$N^2$ will simply be $Y25 \ (100Y+25)$

For example :

$35^2 = ?$

$X=3$
$X(X+1)=3 \times 4={\color{red}12}$

Hence $35^2={\color{red}12}25$

Another example :

$115^2= ?$

$X=11$
$X(X+1)=11 \times 12={\color{red}132}$

Hence $115^2={\color{red}132}25$

colby2152 · #4 $Old$ April 18th, 2008, 02:18 PM

Good hint, but it hardly deals with Calculus concepts... more with arithmetic and algebra!

Moo · #5 $Old$ April 18th, 2008, 02:25 PM

I'm sorry, I don't know that much about the different categories :s

Moo · #6 $Old$ April 18th, 2008, 02:37 PM

Using binomial to avoid long calculations...

If you're asked to multiply two numbers, you can rapidly go through this checking.
See if the numbers are symmetric with respect to a number N.
If so, the numbers a and b can be written as : $a=N-x$ and $b=N+x$
The product $ab$ will be :

$ab=(N-x)(N+x)$

By using a binomial formula, you get :

$ab=N^2-x^2$

Which can sometimes be more sympathical to calculate..

For example :

$58 \times 42 = ?$

Notice that these numbers are symmetric with respect to 50.

$58=50+8$
$42=50-8$

$58 \times 42 =(50+8)(50-8)=50^2-8^2=2500-64=2436$

I know that these methods look elementary, but it sometimes helps when you are in a hurry and not allowed to use any calculators

janvdl · #7 $Old$ April 18th, 2008, 02:51 PM

Moo, I believe the word you are looking for is calculations, not calculus.

Other than that, very nice, good to see someone besides Colby and TPH post some interesting tutorials on things.

Moo · #8 $Old$ April 18th, 2008, 02:54 PM

Quote:

Originally Posted by janvdl $View Post$

Moo, I believe the word you are looking for is calculations, not calculus.

Other than that, very nice, good to see someone besides Colby and TPH post some interesting tutorials on things.

Thanks, I've corrected the mistakes... And I've learnt one more thing !

Moo · #9 $Old$ April 20th, 2008, 05:49 AM

Check for the unit digits (and cubic roots of numbers)

In general cases, when you gotta check rapidly if your calculation seems correct, look at the digit number.

Let $X=x*10+a$ and $Y=y*10+b$

$XY=(ay+x+bx+y)*10+ab$

Hence the unit digit of XY will be the unit digit of ab, which is easier to get.

For example :

$46 \times 73$

Here, $a=6$ and $b=3$ . So $ab=1{\color{blue} 8}$
Hence the unit digit of $46 \times 73$ is ${\color{blue} 8}$
If you find a result whose digit number is not 8, you can directly calculate again.

Here is only a beginning for cubic roots... And shall be known

$1^3={\color{red}1}$
$2^3={\color{red}8}$
$3^3=2{\color{red}7}$
$4^3=6{\color{red}4}$
$5^3=12{\color{red}5}$
$6^3=21{\color{red}6}$
$7^3=34{\color{red}3}$
$8^3=51{\color{red}2}$
$9^3=72{\color{red}9}$
$10^3=100{\color{red}0}$

Notice that each number appears once in the unit digit.

This means that if, for example, you're looking for the cubic root of a number whose unit digit is 6, its cubic root necessarily ends with a 6.

Mathstud28 · #10 $Old$ April 20th, 2008, 08:14 AM

Quote:

Originally Posted by Moo $View Post$

Hello,

1.*1.

This has nothing to do with extraordinary technique, but you gotta think about it (I'm confident that some of you know about it)..

$(10+a)(10+b)=10(10+a)+b(10+a)=10(10+a)+10b+ab=10(10+a+b)+ab$

So multiplying two such numbers is equivalent to :

- sum one of the numbers with the unit digit of the other one
- multiply it by 10
- add the product of the two unit digits.

$19 \times 17= ?$

- $17+9=26$

- $26 \times 10=260$

- $7 \times 9=63$

- $19 \times 17=260+63=323$

I hope this would help at least one person in the future

Enjoy calculations !

Hi Moo,

I was jus wondering if you do this....say I solve an equation and get $x=\frac{288}{128}$ ...my mind works like this $x=\frac{200+88}{100+28}=\frac{4(50+22)}{4(25+7)}=\frac{77}{37}$ that is how my mind simplifies things...its much quicker in my mind though...is that how you process things?

Moo · #11 $Old$ April 20th, 2008, 08:40 AM

Hm...
It was a reflex for me to divide by 4...

I see that 288=280+8 rather than 200+88
Then 120+8 rather than 128..

ice_syncer · #12 $Old$ April 20th, 2008, 08:43 AM

Quote:

Originally Posted by Moo $View Post$

Hello,

1.*1.

This has nothing to do with extraordinary technique, but you gotta think about it (I'm confident that some of you know about it)..

$(10+a)(10+b)=10(10+a)+b(10+a)=10(10+a)+10b+ab=10(10+a+b)+ab$

So multiplying two such numbers is equivalent to :

- sum one of the numbers with the unit digit of the other one
- multiply it by 10
- add the product of the two unit digits.

$19 \times 17= ?$

- $17+9=26$

- $26 \times 10=260$

- $7 \times 9=63$

- $19 \times 17=260+63=323$

I hope this would help at least one person in the future

Enjoy calculations !

well, pretty good, but I knew the mental multiplication tricks, all you have to do is dismantle the multiplication steps and add

Moo · #13 $Old$ April 20th, 2008, 08:45 AM

I know it's a basic thing, but it simplifies steps when you directly apply it