Some times it is necessary to get squares of numbers directly.

Most of us mug up 12 = 1, 22 = 4, 32 = 9, 42 = 16,………..152 = 225 etc

1. Numbers ending with 5. (Have 5 in its unit’s place)

Consider number 25. To find 252

(2/5)2 = 2 ´ 3 / 52 = 6/25 = 625

Divide number 25 into two parts. Units place and the remaining digits as n / 5. For example if n = 4 then square of (4/5) is obtained by placing the product of n = 4 and (n + 1) = 5 as the first part and 52 = 25 as the second part of the answer, which gives 20 / 25.

General formula is (n / 5)2 = n ´ (n + 1) / 52

Using this we get the following squares in one direct step as

1052 = 10 ´ 11 / 25 = 11025

952 = 9 ´ 10 / 25 = 9025

1952 = 19 ´ 20 / 25 = 38025 etc.

2. Yuvadunam Tavdunikritya Vargamch Yojayet:

This sutra means : Increase the number by its deviation from the base and suffix the square of the deviation.

This technique can be used to find squares of number near to the base. Here numbers 10, 100, 1000 etc are called as bases.

Let us see how to final square of 93.

Observe that 93 = 100 – 7. So we say that 93 has a deviation (-7) form the base 100. Now divide the answer in two parts as

number + deviation / (deviation)2.

Remember that the deviation square should have as many digits as many zeros are there in the base.

932 = 93 +(-7) / (-7)2 = 8649 (here base is 100)

1032 = 103 + 3 / 32 = 10609.

In general square of number n with deviation d from the base can be written as n2 = n + d / d2

972 = 97 – 3 / (-3)2 contain as many digits as many digits as

= 9409. many zeros in the base.

(Here 32 = 9, base 100 so me make 9 as 09)

9862 = 986 – 14 / 142 = 972 / 196 = 972196.

10252 = 1025 + 25 / 252 = 1050625

882 = 88 – 12 / 122 = 76 / 144

But base is 100 so 1 becomes carry and is to be added to the first part giving the answer as

= 77 / 44

Check out the time you need to find squares of large numbers with new method. You first ask your friend to do it with usual method and compare the time and strain you both experienced. Is it not amazing that your friend following usual method took longer time, had more chances of mistakes and felt strain where as you did it much farter, more accurate and strain less. Is it not being smarter???

To practice above two methods find

I. 152, 252, 352, 552, 852, 1252, 2052, 1952, 20252

II. 932, 862, 1052, 1122