Rational Numbers, Terminating and Recurring Decimals

Do you remember that we learnt that the ratio of a circles circumference to its diameter is always constant no matter the diameter of the circle? And we learnt that this constant is called pi and the symbol for pi is $\pi$ and that $\pi = 3.14?$

$\pi$ is the circumference ( $c$ ) divided by the diameter ( $d$ ) . $\pi = \frac{c}{d} = 3.14$

This is a simplification. The real value of $\pi$ is 3.14159265…The decimal places keep going forever and have no pattern.

Numbers like $\pi$ with decimal places that keep going forever and have no pattern are called irrational numbers because they cannot be expressed by a ratio of integers like rational numbers can.

Numbers that have decimal places can be terminating such as 0.75. Numbers like these are rational. $0.75 = \frac{3}{4}$ as a fraction and $3:1$ as a ratio.

Numbers can also have an infinite number of decimal places but still be rational. An example is $1:2 = \frac{1}{3} = 0.3333333...=0.\dot{3}$ . The number with the dot above the three is called nought point three recurring because the three keeps on recurring.

If there is more than one dot over decimals then the numbers with the dots above them repeat $0.373737...=0.\dot{3}\dot{7}$ . As a fraction this is $\frac{37}{99}$ and as a ratio 37:62

The dot isn’t always above the first decimal $0.166666...=0.1\dot{6} = \frac{1}{6} = 1:5$

$\sqrt{2}$ is irrational $\sqrt{2} = 1.4142135623...$ $\sqrt{2}$ cannot be expressed as a ratio or a fraction.

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