13th August 2011 -1 Squaring Two Digit Numbers Ending in 5

I saw this method in a GCSE foundation paper, as well as here. Squaring a number is simply multiplying it by itself, so 3 squared, written as 3² means 3 x 3 = 9. That is 3²=9.

Method

This method works for numbers ending in 5. As an example, we'll square 35.

Example	Method
3	1) Write down the tens figure.
3 + 1 = 4	2) Add 1 to it.
3 x 4 = 12	3) Multiply the numbers from steps 1 and 2.
1225	4) Write 25 after the number obtained in step 3. This is the original number squared.

Does this method really work?

Proof

Let the number to be squared be a where a is a number ending in 5. Let the tens digit be b. Then we can write

a = 10b + 5. 

Squaring a gives

a² = (10b + 5)²

= 100b² + 100b + 25. (1)

Now we'll apply the method to a and see if we get the same result. First we write down the tens figure b and then add one to it, giving b + 1. These are then multiplied to give

b(b + 1) = b² + b.

Since, we write 25 immediately after this number, we are actually multiplying by 100, which shifts the 12 in the example above to 1200,  and then adding 25.

100(b² + b) + 25 = 100b² + 100b + 25,

which is what we obtained when we multiplied a by itself in (1). Hence the method gives us a², that is, it works.

Note that this proof can be adapted for any number ending in 5 as we never use that b is a single digit. For instance 115² is found as follows:

11 x 12 = 132     (since 12 = 11+1)
so 115² = 13225 (putting 25 after the number above).