# Math Tricks: Squaring Two-Digit Numbers

November 3rd, 2011 by Math Tricks | Filed under Math Tricks.

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### Squaring Two-Digit Numbers Quickly

Using this math trick, you will be able to square any two digit number very quickly.  With a little practice you can do it in your head, or you can do it on paper and still impress others with your math skills.  This method of squaring is very easy, and I will be using as an example squaring the number 23.

First, determine the closest number to your number that ends in a zero.  In this example, the number is 20.  Next, determine the difference between your number and the closest number with the zero.  In this case, it will be (23 – 20) = 3.  Add the result to your number (23 + 3 = 26).  Now, multiply the number with the zero by the sum of your number and the difference you determined:

20 x 26 = 520

Now square the difference you determined before, and add it to the result above:

(3 x 3) + 520 = 529

Presto!  There is the square of your number!

Here are some more examples:

The Square of 25

Closest zero number:  20 (note – 30 will also work in this example)

Difference:  25 – 20 = 5

Sum:  25 + 5 = 30

Answer:  (5 x 5) + (20 x 30) = 25 + 600 = 625

The Square of 37

Closest zero number:  40

Difference:  37 – 40 = -3

Sum:  37 + -3 = 34

Answer:  (-3 x -3) + (40 x 34) = 9 + 1360 = 1369

The Square of 81

Closest zero number:  80

Difference:  81 – 80 = 1

Sum:  81 + 1 = 82

Answer:  (1 x 1) + (80 x 82) = 1 + 6400 + 160 = 6561

Noticed how I broke down the (80 x 82) into (80 x 80) + (2 x 80)!

The Square of 12

Closest zero number:  10

Difference:  12 – 10 = 2

Sum:  12 + 2 = 14

Answer:  (2 x 2) + (14 x 10) = 4 + 140 = 144

This math trick can sometimes be very useful, and is one that I recommend that you practice on your own.  One day you just might come upon a situation where you need to square a two digit number without a calculator.  I know – you can use your smart phone… but with practice, you can solve the problem before your run your calculator app!

### 5 Responses to “Math Tricks: Squaring Two-Digit Numbers”

1. Mahesh dharmawardane | 28/11/11

(a,b)^2 = a^2,2*a*b,b^2.

Note that the symbols “^” and “*” denote the
power and product, respectively.

Step 1: We first write down b^2. If (b^2)>9,we write the second digit and add the first digit to the result of the Step 2.

Step 2: Now we write down 2*a*b + (first digit of the Step 1 if it is available). If this result is greater than 9, we just write down the second digit here and take the first digit to the Step 3.

Step 3: Finally, we write down b^2 + (the first digit of the result in the Step 2 if it is available).

Example 1: Suppose that we need to find the square of 12. Using the above formula, we write it as follows.

(1,2)^2= 1^2, 2*1*2,2^2= 1,4,4=144.

Example 2:

24^2=(2,4)^2= 2^2,2*2*4,4^2=5,7,6=576

Step 1: 4^2=16. We just keep 6 here and take 1 to
next step.

Step 2: 2*2*4+1=17. We just keep 7 here and take 1
to next step.

Step 3: 2^2+1=5

This can be done for any power of two digits numbers. We can also use this method for any number but it is difficult to calculate by our normal ability.

If you take n is the power then you can just use the binomial expansion for finding the nth power of the two digits number by replacing the “+” with “,”. That is,

(a,b)^n = nC0 a^n,nC1 a^(n-1)b,…, nCn b^n,

where nCr= n!/(r!(n-r)!). “!” denotes the factorial.

Ex: nC0= n!/(0!n!)=1,
nC1=n!/(1!*n-1)!)=n.

2. Vikash | 2/02/12

these process are two long…
i have the new one.

For ex: square of 23.

start from left hand side.
23 (3×3=9)
23 (double the second number and multiply with the first.)
—– (3x2x2=12)
529 (put 2 and carry 1)
(multiply the right side number and add the carry)
(2×2+1=5)

By: Vikash (vkp.jsr@gmail.com)

3. Ernie | 20/04/13

You ought to be a part of a contest for one of the finest sites on the web.
I’m going to highly recommend this web site!

4. Math Tricks | 21/04/13

Very kind of you, Ernie – thanks so much!

5. Utkarsh Bajpai | 6/05/13

Thanks to all of u;
it will be very beneficial for me…

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