Rational Numbers

Rational numbers are one very common type of numbers that we usually study after integers in math. These numbers are in the form of p/q, where p and q can be any integer and q ≠ 0. Most often people find it confusing to differentiate between fractions and rational numbers because of the basic structure of numbers, that is p/q form. Fractions are made up of whole numbers while rational numbers are made up of integers as their numerator and denominator. Let's learn more about rational numbers in this lesson.

Do you know from where the word "Rational" originated? It is originated from the word "ratio". So, rational numbers are very well related to the ratio concept of ratio.

Rational Numbers Definition

A rational number is a number that is of the form p/q where p and q are integers and q is not equal to 0. The set of rational numbers is denoted by Q.

In other words, If a number can be expressed as a fraction where both the numerator and the denominator are integers, the number is a rational number.

Examples of Rational Numbers

If a number can be expressed as a fraction where both the numerator and the denominator are integers, the number is a rational number. Some examples of rational numbers are:

• 1/2
   
• -3/4
   
• 0.3 or 3/10
   
• -0.7 or -7/10
   
• 0.141414... or 14/99

There are different types of rational numbers. We shouldn't assume that only fractions with integers are rational numbers. The different types of rational numbers are:

• integers like -2, 0, 3 etc.
• fractions whose numerators and denominators are integers like 3/7, -6/5, etc.
• terminating decimals like 0.35, 0.7116, 0.9768, etc.
• non-terminating decimals with some repeating patterns (after the decimal point) such as 0.333..., 0.141414..., etc. These are popularly known as non-terminating repeating decimals.

How to Identify Rational Numbers?

In each of the above cases, the number can be expressed as a fraction of integers. Hence, each of these numbers is a rational number. To find whether a given number is a rational number, we can check whether it matches with any of these conditions: 

• We can represent the given number as a fraction of integers
• We the decimal expansion of the number is terminating or non-terminating repeating.
• All whole numbers are rational numbers

Example: Is 0.923076923076923076923076923076... a rational number?

Solution:

The given number has a set of decimals 923076 which is repeating continuously.

Thus, it is a rational number.

Rational Numbers in decimal form

Rational numbers can also be expressed in decimal form. Do you know 1.1 is a rational number? Yes, it is because 1.1 can be written as 1.1= 11/10. Now let's talk about non-terminating decimals such as 0.333..... Since 0.333... can be written as 1/3, therefore it is a rational number. Therefore, non-terminating decimals having repeated numbers after the decimal point are also rational numbers.

Is 0 a Rational Number?

Yes, 0 is a rational number as it can be written as a fraction of integers like 0/1, 0/-2,... etc.

List of Rational Numbers

From the above information, it is clear that there is an infinite number of rational numbers. Hence, it is not possible to determine the list of rational numbers.

Smallest Rational Number

Since we cannot determine the list of rational numbers, we cannot determine the smallest rational number.

Points to Remember:

• Rational numbers are NOT only fractions but any number that can be expressed as fractions.
• Natural numbers, whole numbers, integers, fractions of integers, and terminating decimals are rational numbers.
• Non-terminating decimals with repeating patterns of decimals are also rational numbers.
• If a fraction has a negative sign either to the numerator or to the denominator or in front of the fraction, the fraction is negative. i.e, -a/b = a/-b.

Adding and Subtracting Rational Numbers

The process of adding and subtracting rational numbers can be done in the same way as fractions. To add or subtract any two rational numbers, we make their denominators the same and then add the numerators.

Example : 1/2 - (-2/3)= 1/2 + 2/3 = 1/2 × 3/3 + 2/3 × 2/2 = 2/6 + 4/6 = 6/6 = 1

We can learn more about addition of fractions and subtraction of fractions.

Multiplying and Dividing Rational Numbers

The process of multiplying and dividing rational numbers can be done in the same way as fractions. To multiply any two rational numbers, we multiply their numerators and their denominators separately and simplify the resultant fraction.

Example: 3/5 × -2/7 = (3 × -2)/(5 × 7)= -6/35

To divide any two fractions, we multiply the first fraction (which is dividend) by the reciprocal of the second fraction (which is the divisor).

Example: 3/5 ÷ 2/7=3/5 × 7/2 = 21/10 or \(2\dfrac{1}{10}\)

The numbers which are NOT rational numbers are called irrational numbers. The set of irrational numbers is represented by Q´. The difference between rational and irrational numbers are as follows:

Look at the chart given below to understand the difference between rational numbers and irrational numbers along with other types of numbers pictorially.

What is a Rational Number? Give an Example.

Any number in the form of p/q where p and q are integers and q is not equal to 0 is a rational number. Examples of rational numbers are 1/2, -3/4, 0.3, or 3/10.

How can you Identify a Rational Number?

To identify whether a given number is rational, just convert it into decimal form. If the decimal is terminating or non-terminating with repeating patterns of decimal, then the number is rational. Otherwise, it's irrational.

What are Terminating Numbers?

Terminating numbers are those decimal numbers that end after a specific number of decimal places. For example, 1.5, 3.4, 0.25, etc are terminating numbers. All terminating numbers are rational numbers as they can be written in the form of p/q easily.

What is the Difference Between Rational and Irrational Numbers?

Rational numbers are those that are terminating or non-terminating repeating numbers, while irrational numbers are those that neither terminate nor repeat after a specific number of decimal places.

What are Irrational Numbers?

Irrational numbers are those that cannot be represented using integers in the p/q form. The set of irrational numbers is denoted by Q´.

Is 3.14 a Rational Number?

Yes, 3.14 is a rational number as it is a terminating decimal. But note that π is NOT a rational number because the exact value of π is NOT 22/7. Its value is 3.141592653589793238... which has a decimal but no repeating patterns of decimals.

Is 0 a Rational Number?

Yes, 0 is a rational number as we can write it as 0/1 where 0 and 1 are integers and the denominator is not equal to 0.