FRACTIONS AND DECIMAL NUMBERS 6

Fractional and decimal notations are numerical forms and, as you will now see, many quantities can be expressed in both.

Converting fractions into decimals

We already know that a fraction is an indicated division whose result is a terminating or a recurring decimal.

The irreducible fraction that results in a decimal is called the corresponding fraction of that number.

Corresponding fraction of 1. 2 ! = 9 11 : 9 11 = 11 : 9 = 1. 2 !

Any fraction can be converted into decimal form. To do this, the numerator is divided by the denominator. However, the opposite is not true: we cannot convert all decimals into fractions. We can only convert terminating and recurring decimals into fractions.

Converting terminating decimals into fractions

A terminating decimal can be converted into a fraction by removing the decimal point and dividing it by the unit followed by as many zeros as the original number had decimal places.

Converting recurring decimals into fractions

Look at the examples below which demonstrate the method for finding the corresponding fraction of a pure recurring decimal and a mixed recurring decimal.

• We want to convert 1. 2 ! into a fraction.

Let’s say A = 1.222…

We multiply it by 10 and subtract the number itself:

• We want to convert 0.7 2 ! into a fraction.

Let’s say B = 0.7222…

We multiply it by 100 and subtract 10 times the number itself:

Rational numbers

• We say that a number is rational when we can express it in fractional form.

rational number = Integer Integer

• The set of rational numbers is represented by the letter Q

• A rational number can be expressed in many different ways. For example: 0.2 = 10 2 5 1 15 3 20 4 250 50 = === = = …

• All integers and, therefore, all natural numbers are rational. So, any integer can be expressed as a fraction: 3 = 1 3 2 6 = = … –3 = 1 3 2 6 – –=

• Terminating and recurring decimals are also rational numbers. As we have seen, these numbers can always be expressed in a fractional form: 0.35 = 20 7 . 14 9 13 = ! . 02 15 330 71 = #

• Decimals with infinite, non-recurring decimal places are not rational. For example: 2 = 1.414213… π = 3.141592… 5 = 2.236067… These numbers cannot be expressed in fractional form. All the above can be summarised in the following diagram: