Real Numbers and Their Decimal Expansions

In Mathematics, a real number is a combination of rational numbers and irrational numbers. The rational and irrational numbers can be expressed in their decimal form. The real numbers can be represented using the number line. If the numbers cannot be expressed on the number line, then the numbers are called imaginary numbers. In this article, we are going to learn the real numbers and their decimal expansions in detail.

Table of Contents:

• What are Rational and Irrational Numbers?
• Decimal Expansion of Real Numbers
  • Terminating Decimal
  • Non-terminating and Repeating Decimal
  • Non-terminating and Non-repeating Decimals
• Solved Examples
• Practice Problems
• FAQs

What are Rational and Irrational Numbers?

The rational numbers are the numbers that can be expressed in the form of p/q, where q ≠ 0. If a number cannot be expressed in the form of p/q, then the numbers are called irrational numbers.

Examples of Rational Numbers: 2/5, 9/3, 24/4, and so on.

Examples of Irrational Numbers: √2, √3, π, and so on.

Decimal Expansion of Real Numbers

We know that the combination of rational and irrational numbers is called real numbers. Now, we will have a look at the decimal expansion of rational and irrational numbers. The decimal expansion of real numbers can be classified into three types. They are:

• Terminating Decimals
• Non-terminating and Repeating Decimals
• Non-terminating and Non-repeating Decimals

Terminating Decimals

The decimal expansion terminates or ends after finite numbers of steps. Such types of decimal expansion are called terminating decimals. It means that, after the decimal point, the numbers come to an end at a certain point.

For example, ½ is a rational number and its decimal expansion is 0.5.

Note: Terminating decimals are rational numbers.

Non-terminating and Repeating Decimals

In non-terminating decimals, the decimal expansion does not come to an end and it has an infinite number of digits. The repeating decimals are the decimals, where a certain number of digits uniformly repeats after the decimal point.

An example of non-terminating and repeating decimals is 1.454545…Here, the digit 45 constantly repeats after the decimal point.

Note: Non-terminating and repeating decimals are rational numbers.

Non-terminating and Non-repeating Decimals

Non-terminating and non-repeating decimals are one of the types of decimal expansion, in which the number after the decimal point is non-terminating and the decimal numbers are not repeating.

An example of non-terminating and non-terminating decimal is 2.34765….. Here, the numbers after the decimal point are infinite and they are not repeating.

Note: Non-terminating and non-repeating numbers are irrational numbers.

Video Lesson on Numbers

Examples

Go through the following examples to understand the concept of real numbers and their decimal expansion.

Example 1:

Prove that 3.1426 is a rational number.

Solution:

To prove: 3.1426 is a rational number.

The number 3.1426 can be written as 31426/10000

31426/10000 = 3.1426

Hence, the number 3.1426 can be expressed in the form of 31426/10000, which is equal to p/q, where q≠ 0.

Hence, proved.

Example 2:

Show that the decimal expansion of 14/11 is non-terminating and repeating.

Solution:

Given number: 14/11

The decimal expansion of 14/11 is 1.2727272727…

Here, the number 27 after the decimal point is repeating and the number 1.2727272727…is non-terminating.

Since the given number 14/11 is non-terminating and repeating, we can say that 14/11 is a rational number.

Hence, proved.

Also, read: Operations on Real Numbers.

Practice Problems

Solve the following problems:

1. Prove that 0.2353535 is a rational number.
2. Convert the following real numbers in their decimal expansion:
   1. 1/11
   2. 329/400
   3. 3/13
3. Classify the following numbers as rational or irrational numbers:
   1. √225
   2. 7.478478…