Rational Numbers

Rational numbers are in the form of p/q, where p and q can be any integer and q ≠ 0. This means that rational numbers include natural numbers, whole numbers, integers, fractions of integers, and decimals (terminating decimals and recurring decimals). 

The word ‘rational’ originated from the word ‘ratio’. So, rational numbers are well related to the concept of fractions which represent ratios. 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.

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. Observe the following figure which defines a rational number.

Examples 

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 as follows.

• 56 (which can be written as 56/1)
• 0 (which is another form of 0/1)
• 1/2
• √16 which is equal to 4
• -3/4
• 0.3 or 3/10
• -0.7 or -7/10
• 0.141414… or 14/99

How to identify rational numbers?

Rational numbers can be easily identified with the help of the following characteristics.

• All integers, whole numbers, natural numbers, and fractions with integers are rational numbers.
• If the decimal form of the number is terminating or recurring as in the case of 5.6 or 2.141414, we know that they are rational numbers.
• In case, the decimals seem to be never-ending or non-recurring, then these are called irrational numbers. As in the case of √5 which is equal to 2.236067977499789696409173… which is an irrational number.
• Another way to identify rational numbers is to see if the number can be expressed in the form p/q where p and q are integers and q is not equal to 0.

The set of rational numerals:

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

Example: Is 0.923076923076923076923076923076… a rational number?

Solution: The given number has a set of decimals 923076 which is recurring and repeated continuously. Thus, it is a rational number.

Let us take another example.

Example: Is √2 a rational number?

Solution: If we write the decimal value of √2 we get √2 = 1.414213562….which is a non-terminating and non-recurring decimal. Therefore, this is not a rational number. It is an irrational number.

The different types of rational numbers are given as follows.

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

Standard Form of Rational Numbers

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.

For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number ⅓ is in standard form.

Positive and Negative Rational Numbers

As we know that the rational number is in the form of p/q, where p and q are integers. Also, q should be a non-zero integer. The rational number can be either positive or negative. If the rational number is positive, both p and q are positive integers. If the rational number takes the form -(p/q), then either p or q takes the negative value.

It means that

-(p/q) = (-p)/q = p/(-q).

Now, let’s discuss some of the examples of positive and negative rational numbers.