Squaring numbers ending in 9

Here are three techniques to square any number ending in 9. All the techniques can be applied to any number, no matter how many digits it has, so long as the unit’s digit is 9. Of course, as the number gets larger, multiplying the two numbers of the first step will become cumbersome unless a short cut can be used. However, most two- and three digit numbers ending in 9 can be readily squared, once a facility with the other short-cut methods has been achieved.

Technique-1

To square a number ending in 9; e.g. 29, 59, 129...., first square the next number that is ending in zero, and then subtract both the number and the number next to it.

29² =?

The number to be squared is 29 and the number next to it is 30. Finding square of 30 is simple as it ends in ‘0’.

So, 29² = 30² - (29 + 30) = 900 - 59 = 841

So, 79²=?

The number to be squared is 79 and the number next to it is 80.

So, 79² = 80² - (79 + 80) = 6400 - 159 = 6241

Further example, 129² =?

129² = 130² - (129 + 130) = 16900 - 259 = 16641

Technique-2

Step-1: Write down 1.

Step-2: Add 1 to the ten’s digit and multiply by 8. Write this number down, carry if necessary.

Step-3: Multiply the number in the ten’s digit by that number plus 1. Write this result.

Example: 69² =?

1. Write down 1.

2. 8 X (6 + 1) = 56. Write 6, carry *5.

3. 6 X (6 + 1) = 42 + *5 = 47. Write 47.

The answer is 4761.

Another example: 249² =?

1. Write down 1.

2. 8 X (24 + 1) = 200. Write 0, carry *20 (see ‘Multiplying by 25’).

3. 24 X (24 + 1) = 600 + *20 = 620. Write 620.

The answer is 62001.

Technique-3

Multiply the number to the left of the 9 by two more than itself. Affix an 8 to the result and subtract twice the number to the left of the 9. Affix a 1 to the result.

Example: 79²=?

The number to the left of the 9 is 7. Two more than this is 9. Multiply 7 by 9.

7 x 9 = 63

Affix 8 to get 638.

Subtract twice the number to the left of the 9,

638 - (2 x 7) = 638 - 14 = 624

Affix a 1 to obtain the final answer 6,241.

So, 79² = 6,241

Example: 249²=?

The number to the left of the 9 is 24. Two more than this is 26. Multiply 24 by 26. (Multiplication short cut method can be used).

24 x 26 = 625 - 1 = 624

Affix 8.