Rational Numbers

What is a Rational Number?

In the world of Mathematics, all the digits are arranged systematically under different categories, depending on the type and characteristics. Normally, the categories begin from highest to lowest which are: complex numbers, real numbers, rational numbers, integers, whole numbers and the natural numbers. There exist many numbers that come under more than one category.

Rational numbers can be written in the form of p/q, where \(q\neq 0\) ie. in terms of simple fractions.

The set of rational numbers –

  1. Involve positive and negative numbers, and zero.
  2. Can be expressed as a fraction.

The name rational lies on the word ‘ratio’ which is a comparison of two or more values and generally said as a fraction. A number is said to be a rational number if you can write it as one integer divided by the other integer.

Irrational Numbers

All the numbers that are not rational are called as irrational. An irrational number can be written as a decimal, but not the fraction. It has endless non-repeating digits to the right side of the decimal point. Some of the irrational numbers are mentioned below.

\(\pi\) = 3.142857…

√2 = 1.414213…

Rational Numbers and Irrational Numbers

Rational Numbers and Irrational Numbers

Although irrational numbers are not used much, they have their existence on the number line. In fact, there is an infinite number of irrational numbers between 0 & 1 on the number line.

The number ½ is a rational number because it is read as integer 1 divided by the integer 2. You can also consider 15/3 or 10/5 as rational numbers since 15 divided by 3 equals to 5 and 10/5 equals to 2. The numbers mixed such as 1½ also comes under rational number as we can write it as 3/2.

Rational numbers can be either positive, negative or zero. While specifying a negative rational number, you put the negative sign either in front or with the numerator and that is the standard mathematical notation. For example, we denote negative of 5/2 as -5/2.

Rational Numbers Examples

Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both the numerator and the denominator are whole numbers.

  • 6 is a rational number since it can be written as the fraction 6/1.
  • Similarly, ¾ is also a rational number as it can also be expressed as a fraction.
  • Even big complex fraction like 907/56, 008 is rational, because it can also be written as a fraction.

Rational Numbers

Rational numbers (R)include all the real numbers (Q).

Real numbers include the integers (Z).

Integers involves the natural numbers(N).

Every whole number is a rational number because every whole number can be expressed as a fraction.

Infinite steps of division.

Non-terminating decimals:

These are decimals where numbers after decimal recur or repeat.

Example 1 : Check whether decimal expansion 15 divided by 7 is terminating or non-terminating decimal.

Solution: Divide 15 by 7

Rational Numbers

15/7=2.142857142857143

Here, the digits after decimal are recurring. Hence, decimal expansion of 15/7  is a non-terminating decimal.

Example 2: Represent 3/6  as a decimal.

Solution: Divide 3 by 6

Rational Numbers

3/6=0.5

Example 3: Express 5/13   in decimal form.

Solution:

Rational Numbers

5/13=0.3846153846153846

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Practise This Question

The decimal number 0.625 when expressed as a fraction is