The following algebraic proof for squaring two digit numbers is presented for the general case of multiplying any two digit numbers as well as a proof for each of the special cases; squaring two digit numbers that end with five and squaring two digit numbers that begin with five. These two can be considered shortcuts of the general method.

#### Squaring Any Two Digit Numbers

A two digit number, say 46 can also be represented as .

We can generalize this to use letters instead of numbers:

So if we had the square of it would be which we can also represent as

Which we can expand out to

Which is the rule for squaring two digit numbers.

1 - square the units digit

2 - multiply the two digits together and double

3 - square the tens digit

#### Squaring Two Digit Numbers Ending In Five

For the shortcut where the two digit number ends in 5 we can represent the number as:

The square this we can represent as

Which we can expand out to

Which is our shortcut when squaring two digit numbers ending in 5 where we take the tens digit and multiply it by the next higher digit and then use the result as the left hand side of the answer where the right hand side is 25.

#### Squaring Two Digit Numbers Beginning With Five

For the shortcut where the two digit number starts with 5 we can represent the number as:

The square this we can represent as

Which we can expand out to

Which is our shortcut when the numbers start with 5.We take the square of the units digit, as a two digit result, with a leading zero if required as the right hand side of our answer. Then we add the units digit to 25 to get the left hand side of the answer.