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Investigate the concept of irrational numbers, including π (ACMNA186)

Source: Australian Curriculum, Assessment and Reporting Authority (ACARA)


Source: Australian Curriculum, Assessment and Reporting Authority (ACARA)

Irrational numbers

The number \(\sqrt{2}\) arises as the hypotenuse of a right-angled triangle of side length 1.
The Greeks realized that this number was not rational, which means that it could not be written as a fraction \(\dfrac{a}{b}, b \neq 0.\)

There are many proofs of this, mostly involving contradiction, and here is one.

We first prove the following result about fractions:


If a fraction \(\dfrac{b}{a}\) is not a whole number, then its square \(\dfrac{b^2}{a^2}\) is not a whole number either.


Suppose that \(\dfrac{b}{a}\) is in reduced form, so that a and b have no common factors except 1. Then \(a^2\) and \(b^2\) also have no common factors, because if any prime p were a common factor of \(a^2\) and \(b^2\), it would also be a common factor of a and b. 
Hence \(\dfrac{b^2}{a^2}\) is not a whole number.

Proof that \(\sqrt{2}\) is irrational

Since \(\sqrt{2}\) is not a whole number, but its square is the whole number 2, it follows from the above result that \(\sqrt{2}\) is not a rational number.

This caused great upset to the Greek mathematicians, since it introduced a new sort of number which they called 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.

Numbers such as \(\sqrt{2},\) \(\sqrt{7},\) \(\sqrt{29}\) are called surds.

In addition, numbers such as π, log\(_{10}3\), log\(_23\), sin 22°, and so on, are also irrational. There is no general method for telling when a number is irrational, and indeed there are numbers such as π + e that arise in mathematics whose status is currently unknown.