The Square Numbers

The square numbers are generated by multiplying integers by themselves (aka “squaring”). Given that multiplying 2 numbers with the same sign always yields a positive product, it follows that square numbers are always positive, whether they are the result of squaring a negative or positive integer, so, for example, the squares of both fifteen and negative fifteen is two hundred and twenty five.

Since even the natural numbers form an infinite set, it follows that the squares also form an infinite set.

One way to define a square number is to consider its factors, and in particular its factor pairs. Since one of the factor pairs of any square number is a repeated factor, it follows that all square numbers have an odd number of distinct (positive) factors, whereas all non square  numbers have an even number of distinct factors.

As we progress through the square numbers they grow further and further apart, alternate between odd and even numbers and are progressively separated by the odd numbers.

Perhaps the most common application of the squares we meet in high school maths is their use in Pythagoras’ Theorem which gives the relationship between the three sides of right angle triangles. We shall explore this in episode 5.



nrich activities with square numbers