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Supporting Australian Mathematics Project: Middle years.

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Year 7

Number and Algebra

Square roots and square numbers

Factoring whole numbers into products of prime numbers gives rise to some interesting observations. It is a simple way to find the highest common factor between two numbers, and in this module, it allows one to decide if a number is a square number or not. We use the term square to generally mean a common and widely used shape, the regular polygon of four sides. A perfect square is a number that is a product of a whole number with itself. The whole numbers can be divided into numbers that are perfect squares, and numbers that are not perfect squares. The number which is multiplied by itself is called the square root of the perfect square.

Obviously, not all whole numbers are perfect squares. The ancient Greeks knew that the diagonal of a square of side length 1 unit was not an integer, and not a rational number. But \(\sqrt{2}\) squared is two, a whole number. This caused great upset to the Greek mathematicians, since it introduced a new sort of number which we now call an irrational number. An older word for this is 'incommensurable', which meant that it could not be measured as a ratio of two whole numbers. This discovery caused a dramatic rethink into the nature of number. The validity of many of their geometric proofs, which assumed that all lengths could be measured as ratios of whole numbers, was also called into question.