Decimal Numbers

Decimal Numbers

All integers and fractions with integer numerators and non-zero integer denominators are rational numbers. Remember that when the denominator of a fraction is zero then the fraction is undefined.

You can write any rational number as a decimal number but not all decimal numbers are rational numbers. These types of decimal numbers are rational numbers:

• Decimal numbers that end (or terminate). For example, the fraction \(\cfrac{4}{10}\) can be written as \(\text{0.4}\).

• Decimal numbers that have a repeating single digit. For example, the fraction \(\cfrac{1}{3}\) can be written as \(\text{0.}\dot{3}\) or \(\text{0.}\overline{3}\). The dot and bar notations are equivalent and both represent recurring \(\text{3}\)'s, i.e. \(\text{0.}\dot{3} = \text{0.}\overline{3} = \text{0.333...}\).

• Decimal numbers that have a recurring pattern of multiple digits. For example, the fraction \(\cfrac{2}{11}\) can also be written as \(\text{0.}\overline{18}\). The bar represents a recurring pattern of \(\text{1}\)'s and \(\text{8}\)'s, i.e. \(\text{0.}\overline{18} = \text{0.181818...}\).

Notation: You can use a dot or a bar over the repeated digits to indicate that the decimal is a recurring decimal. If the bar covers more than one digit, then all numbers beneath the bar are recurring.

If you are asked to identify whether a number is rational or irrational, first write the number in decimal form. If the number terminates then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational.

When you write irrational numbers in decimal form, you may continue writing them for many, many decimal places. However, this is not convenient and it is often necessary to round off.