Square a 2 digit number faster than a calculator | Part -1 | (Base Method)

Have you ever wondered how some people square numbers faster than a calculator! Guess what! You can do that too! Today, in this post, I am gonna teach you how to square 2 digits number faster than other people could on a calculator. It's gonna take some practice but believe me after some practice you'll be able to square 2 digit numbers in just 2 or 3 seconds max.

(See also:  2 Mental Math Books You Need To Read Right Now!)

The Recipe

The only prerequisite for this method is, knowing the squares of numbers from 1 to 25.

For numbers 26-75

Let's say we want to find 

52² 

1. Subtract 50 from the number. (52-50) = 2.
2. Add the difference to 25. (25+2) = 27. This step gives us the left side of the answer. 
3. Square the difference and that becomes the right side of the answer. 2² = 4.
4. 25+2/2² = 27/04 =2704. 

The '/' represents the two sides of the answer. Both the sides need to have 2 digits each. That's why we added a zero with 2.

Let's do some more examples to clear any doubts you might be having.

59²

1. Subtract 50 from the number. (59-50) = 9.
2. Add the difference to 25. (25+9) = 34.
3. Square the difference and that becomes the right side of the answer. 9²=81.
4. 25+9/9²= 34/81 =3481

Let's practice some numbers smaller than 50.

49²

1. Subtract 50 from the number. (49-50) = -1
2. Add the difference to 25. (25+(-1)) = 24
3. Square the difference and that becomes the right side of the answer. -1²=1
4. 25-1/-1² = 24/01 = 2401

41²

1. Subtract 50 from the number. (41-50) = -9
2. Add the difference to 25. (25+(-9)) = 16
3. Square the difference and that becomes the right side of the answer. -9²=81
4. 25-9/-9² = 16/81 = 1681

Let's practice some numbers where the square of the difference is a 3 digit number.

61²

1. Subtract 50 from the number. (61-50) = 11
2. Add the difference to 25. (25+(11)) = 36
3. Square the difference and that becomes the right side of the answer. 11²=121
4. 25+11/11² = 36/121 
5. Now the answer absolutely can't be 36121. Always remember, the right side can have only 2 digits! So we'll carry forward the 1 and add it to 6.
6. 36+1/21 = 3721

69²

1. Subtract 50 from the number. (69-50) = 19
2. Add the difference to 25. (25+(19)) = 44
3. Square the difference and that becomes the right side of the answer. 19²=361
4. 25+19/19² = 44/361 
5. Now the answer absolutely can't be 44361. Always remember, the right side can have only 2 digits! So we'll carry forward the 3 and add it to 4.
6. 44+3/61 = 4761

39²

1. Subtract 50 from the number. (39-50) = -11
2. Add the difference to 25. (25+(-11)) = 25-11 = 14
3. Square the difference and that becomes the right side of the answer. -11²=121
4. 25-11/11² = 14/121 
5. Now the answer absolutely can't be 14121. Always remember, the right side can have only 2 digits! So we'll carry forward the 1 and add it to 1.
6. 14+1/21 = 1521

For Numbers 75-100

To find out

91²

1. Subtract The Number from 100. (100-91) = 9.
2. Subtract the difference from the number. (91-9) = 82.
3. Square the difference and that becomes the right part of the number. 9²=81.
4. 91-9/9² = 82/81 = 8281.

98²

1. 100-98=2.
2. (98-2)=96
3. 2² = 04
4. 98-2/2² = 96/04 = 9604

89²

1. 100-89 = 11
2. 89-11 = 78
3. 11² = 121
4. 89-11/11² = 78/121
5. 78+1/21 = 79/21 = 7921

If you have some difficulties by this method, or if you don't know the squares till 25 by heart, then be sure to check out this method, Squaring Numbers Faster than Calculator | Part-2 | x^2 - y^2 method, which only requires you to learn squares till 9.

How this Trick Works?

For numbers 25-75.

It's just a simple identity we have all used. (a+b)²

We know, (a+b)² = a² + b² + 2ab.

Now, if we see this.

(50+x)² = 50² + x² + 2*50*x

=2500+x²+100x

Now let's rearrange it. 

2500

+x²

+100x

--------

25+x/x²

-----------

So that's how this trick works. Let's try putting a value to x.

54² = (50+4)²

2500

+16

+400

--------

25+4/16

----------

=> 2916

It is just the same for numbers 75-100

Let the original number be y, So,

y = 100-x, ...(1)

y² = (100-x)² = 100² + x² - 100*x*2

Let's rearrange it.

10000

+x²

-200x

--------

100-2x/x²

---------

We can also write

100-2x as 100-x-x.

By Eq 1 we can say,

100-2x = y-x

That's why we write the square of the number as,

y-x/x²

Let's try substituting a value for x.

a (99)² = (100-1)²

=> 100² + 1 - 100*2*1

Let's arrange it.

10000

+1

-200

---------

10-2/01

---------

10-2 can be written as 9-1.

Therefore, we can write, (99)² as 99-1/1² => 98/01 => 9801.

Bonus

You can also use this method to square numbers from 100-125. Instead of subtracting the difference from the number, just add it. For eg, if we want to square (102), We can just add 2 to 102 and square it like other numbers. Therefore, the 102² would be 102+2/2² => 104/04 => 10404.

Here, we squared numbers near 50 and 100, we can also square numbers near 1000 and 500, or be that case, 10000 and 5000. 

Let's try, 

1024² 

1. (1024-1000) = 24
2. 1024+24/24²
3. 1048/576 = 1048576
4. While squaring numbers near 100, we could only have 2 digits in the right part of the answer because 100 has 2 zeroes. The number of zeroes is equal to the number of digits you can have on the right side of the answer.

998²

1. (1000-998) = 2
2. 998-2/2²
3. 996/004 => 996004

525²

1. (525-500) = 25
2. 250+25/25²
3. 275/625 => 275625
4. Notice, here we subtracted the number from 500 and not 50, because we took 500 as the base this time, and let's see who can answer why added the difference to 250 and not 25, Let me know your answers in the comments down below.

I hope this post was worth your time. Let me know if you liked it, and if you found my efforts worth it, then please share with your friends and help this blog to grow ^_ ^.