Rational Numbers

What are rational numbers?

A rational number is the number of the form p/q, where p and q are integers. They consist of natural numbers, whole numbers, integers, a fraction of integers and decimals. Most of the numbers that we use are rational numbers. Every integer, whole numbers, natural numbers and fractions are rational numbers. But not all rational numbers are integers, whole numbers, natural numbers and fractions.

Let us take an example:
7.5 – This is a rational number as it can be expressed in the p/q form
7.5 = 75/10
= 7 1/2

The word rational is derived from the word “ratio” and therefore rational numbers are related to the concept of fractions that depict ratios. Simply put, a number is a rational number when it is expressed in the p/q form where, q 0 and both the numerator and denominator are integers.

The formula for rational numbers

Any number that can be expressed by way of p/q, where q is not equal to zero is a rational number. Further p/q can be simplified to give decimal numbers.

Representation of rational numbers

Let us look at some rational numbers examples.

    46 – since it can be expressed as 46/1
    0 – which is another form of 0/1
    √4 = 2
    0.2 or 2/10
Integers, Natual, Whole and Rational numbers
Fraction form
Rational or Irrational?

Types of rational numbers

There are different types of rational number as listed below:

Integers like -2, 0, and 5 are rational numbers

Fractions where both the numerator and denominator are whole numbers like 2/7, -2/3 etc. are rational numbers.

Terminating decimals like 0.35, 0.667, 0.9768 etc.

Non-terminating decimals with repetitive patterns 0.222…., 0.141414……etc. are rational numbers. These are commonly known as non-terminating recurring decimals.

Positive rational numbers

Those rational numbers both the numerator and denominator are positive integers are called positive rational numbers. These are greater than 0.
For example, 12/17, 26/11 etc.

Negative rational numbers

Those rational numbers where either the numerator or the denominator have a negative sign, it is referred to as a negative rational number. All negative rational numbers are less than 0.
Examples: -3, -4, -5, -2/3 etc.

Expressing rational numbers in different form

Rational numbers in different forms

Arithmetic Functions of rational numbers

For adding and subtracting rational numbers the same rules of addition and subtraction of integers apply. Let us look at an example.

Subtraction and Addition of Rational Numbers

1. Add two rational numbers with equal denominators.
Let us take the following example: 1/2 + (-5/2)

Adding the numerators
Arithmetic operation on rational numbers

2. Add two rational numbers with different denominators.
Find the LCM of the denominators and convert each of the rational numbers into the equivalent forms by keeping the denominators as of the LCM. Since the equivalent forms have the same denominators, now use the previous method to add them.

Let us consider 2/3 + 5/4

Addition of rational numbers with different denominators

3. Subtract two rational numbers with equal denominators.
Example: 7/3 – 5/3

Addition of numerators

4. Subtract two rational numbers with different denominators.
We begin by simplifying 1/2 – (-2/3).

For that, we will follow the rule of addition and subtraction of numbers which states that two negatives make a positive and the sign of the subtrahend gets reversed. This will make it 1/2 + 2/3

Now, we need to add these fractions 1/2 + 2/3

Subtraction of rational numbers

Further by using the rules of addition of fractions, we will convert the denominator into a common one by finding their lowest common multiple so that it becomes easier to add them.

Therefore, we need to find the LCM of the denominators 2 and 3 which is 6. Then we will convert the fractions to their corresponding equivalent fractions which will make them 3/6 + 4/6. This will give the sum as 7/6 which can be written in the form of a mixed fraction of 1/1/6.

Multiplication and division of Rational Numbers

The multiplication and division of rational numbers can be done in a similar way as that of fractions. First, we multiply their numerator and denominator and then simplify it to get the final answer

For example, let’s multiply 2/5 * 3/2

Addition of rational numbers

First, we will need to multiply the numerator in this case it is 2 * 3 = 6; now we will multiply the denominator 5 * 2 = 10

Multiplication of numerators and denominators

Finally, we have 6/10 that can be simplified further as written as 3/5


Let’s divide 4/6 ÷ 3/8

Multiply numerator and denominator

First, we will need to note the reverse of the second fraction and multiply it with the 1st one. 4/6 * 8/3

Create reciprocal of rational number

Now we will multiply the numerator 4 * 8 = 32; and then the denominator 6 * 3 = 18.

Multiply numerator and denominator

Finally the outcome is 32/18, which when simplified gives 16/9 that can be written as 1.7.

Rational and Irrational numbers

Rational numbers are fractions that don’t have a 0 as a denominator. E.g., 1/2. All numbers that are not rational numbers are irrational. E.g., π (pi)

Rational numbers consist of terminating decimals. They can also be non-terminating decimals with repetitive patterns. E.g.,1.212, 212. While irrational numbers don’t have terminating decimals or have an accurate value. They also don’t have a repetitive decimal pattern. The sum of rational and irrational numbers is an irrational number.

Difference between Rational and Irrational numbers

Rational NumbersIrrational Numbers
Rational numbers are the ones that can be expressed as a ratio of two numbers p & q, where p & q are any integers and where q is not equal to 0.Irrational numbers are those that can not be expressed as a ratio of two number p & q
These numbers are finite or recurring in nature.These are non-terminating and non-repeating in nature.
Rational numbers can be written in fraction form in which both - numerator & denominator are integers and the denominator is not equal to zero.Irrational numbers cannot be expressed in the form of a fraction.
Rational numbers include perfect squares like 4, 9, 16, 25……..Irrational numbers, on the other hand, include surds such as √2, √3, √5 and so on.

Is 0 a rational number?

Zero is a rational number as it can be written as 0/1 (p/q form) where q could be non-zero.

Is Pi a Rational number?

Pi is not a rational number as its exact value is 3.141592653589793238. This is a non-terminating and non-recurring number. When we round off the value of pi to 3.14 then it becomes a rational number since it becomes terminating.

Can the denominator of a Rational number be zero?

Denominator of a Rational number can never be zero.

Did you know?
Although it is a known fact that rational numbers evolved under the Greeks. But the research dates back to 4000 years ago with Babylonians. They coped with fractions back then. The concept evolved further with Egyptians and then with Greeks like Pythagoras, Euclid, Eudoxus and others.