Decimal Expansion of Rational Numbers

The reading material provided on this page for Decimal Expansion of Rational Numbers is specifically designed for students in grades 7 to 12.

Rational Numbers

Rational numbers are numbers that can be expressed in the form of p/q form where p and q are integers and q ≠ 0. Rational numbers can also be expressed by decimal expansion. To understand rational numbers better we need to understand the decimal expansion of rational numbers.

Decimal Expansion

A decimal expansion is a way of writing a number using a decimal point and digits after the decimal point.

For example, the decimal expansion of 1/2 is 0.5.

Rational number by decimal expansion: A number is rational if and only if its decimal expansion is either terminating or non-terminating and repeating.

Irrational number by decimal expansion: A real number is irrational if and only if its decimal expansion is non-terminating and non-repeating.

Types of Decimal Expansion

There are two types of decimal expansion.

Terminating

A terminating decimal representation is a decimal number that ends after a finite number of digits.

Example: Let’s take the fraction 3/5, it terminates after one digit.
3/5 = 0.6
Here 0.6 is the terminating decimal expansion.

Non-terminating

A non-terminating decimal expansion is a decimal number that goes on indefinitely after the decimal point.

1. Non-terminating and repeating (recurring): A decimal number that has a repeating pattern of digits after the decimal point.

For example, the decimal expansion of 1/3 is a non-terminating and recurring decimal: 0.33333..., where the digit 3 repeats infinitely.

2. Non-terminating and non-recurring: A decimal number that does not have a repeating pattern of digits after the decimal point.

For example, the decimal expansion of pi(π) is a non-terminating, non-repeating decimal: 3.1415926535897932384626433832795..., which goes on infinitely without repeating a part.

Conversion of Non Terminating and Recurring decimals in p/q form

To convert a non-terminating and recurring decimal number into a fraction in p/q form, you need to follow these steps:

Step 1: Let x be the non-terminating decimal number.

Step 2: Multiply both sides of the equation by 10ⁿ, where n is the number of digits after the decimal point that you want to keep. This will eliminate the non-terminating decimal and convert it into a terminating decimal.

Step 3: Let y be the resulting terminating decimal.

Step 4: Write y as a fraction in p/q form.

Step 5: Simplify the fraction to the lowest terms if possible.

Example: Convert 0.777... to a fraction in p/q form.

Solution: Let x = 0.7... (1)
Multiply both sides by 10 (because there is one repeated digit after the decimal point that we want to keep)

10x = 7.7... (2)
By subtracting equation (1) from equation (2)
10x – x = (7.7…) – (0.7…)
9x = 7
x = 7/9

Therefore, 0.777... is equivalent to the fraction 7/9.

The above example shows that any non-terminating, repeating decimal number can be represented as a fraction in the form of p/q, where p and q are integers and q ≠ 0. This implies that non-terminating, repeating decimals are rational numbers.