*Sanjay Guilati (author) has been teaching computer and mathematics in Bhilai , state Chhattisgarh in India for 15 years. Currently he is a teacher in a seniorsecondary school and he is also involved in teacher training. Sanjay is the writer of the new blog titled Mathematics Academy.*

*****

**Vedic Mathematics 2 – Squaring Numbers Ending in 5**

*by Sanjay Gulati*

Square Numbers Ending in 5

* *

By way of notation, 2^{2} means 2 times 2 or 4 and 3^{2} means 3 times 3 or 9. What if you are asked to calculate 45^{2} or 65^{2 }or the square of any number ending in a 5 ( a number with unit digit 5) ? More than likely, we would need pencil and paper or a calculator to work it out. One of the Vedic Mathematics Sutras “one more than the one before” can help us to solve the questions immediately. Very simply, multiply the digit(s) to the left of the 5 by the next higher number and then write 25 after it.

For example, let’s apply this sutra to the calculation of 45^{2}.

**1 ^{st} step**: Determine the number to the left of the 5. That number is 4 in our example.

**2 ^{nd} step**: Multiply this number by next higher number . This means multiply the 4 by 5. This results in the number 20.

**3 ^{rd} step**: Follow this result with the number 25. This means the number 25 will follow 20. That is 2,025.This is the answer to the problem.

Let’s try 65^{2}:

**1 ^{st} step:** The number 6

**2**6 times 7 equals 42

^{nd}step:**3**follow 42 with 25, which is 4225.

^{rd}step:**Algebraic proof:**

Consider (ax + b)^{2} = a^{2}. x^{2 }+ 2abx + b^{2}.

This identity for x = 10 and b = 5 becomes

(10*a* + 5)^{ 2 }= *a*^{2 }. 10^{2} + 2. 10*a* . 5 + 5^{2 }

= *a*^{2 }. 10^{2} + *a*. 10^{2} + 5^{2}

= (*a*^{ 2}+ *a* ) . 10^{2 }+ 5^{2 }

= *a* (*a* + 1) . 10^{ 2 }+ 25

Clearly 10*a* + 5 represents two-digit numbers 15, 25, 35,…,95 for the values a = 1, 2, 3, … ,9 respectively. In such a case the number (10*a* + 5)^{2 }is of the form whose L.H.S is ** a (a + 1)** and R.H.S is 25, that is,

*a*(*a*+ 1) / 25*.** **

Similar proof can be given for 3 or more digit numbers.

*The / symbol denotes that the numbers *a*(*a*+1) and 25 were concatenated.