Rational Numbers - Definition, Standard Form, Properties & Questions

The reading material provided on this page for Rational Numbers and its Properties is specifically designed for students in grades 7 to 12. So, let's begin!

What is a Rational Number?

A rational number is a number that can be expressed as the ratio of two integers, where the denominator is not equal to zero. When a rational number (i.e., fraction) is divided, the resulting quotient may be expressed as a decimal, which can be classified as either a terminating decimal or a repeating decimal.

Examples of Rational Numbers

-> 1/2 (half)
-> 3/4 (three-fourths)
-> 7/5 (seven-fifths)
-> -4/3 (negative four-thirds)

Symbol for Rational Numbers

The symbol for rational numbers is "Q".

Standard Form of a Rational Number

The standard form of a rational number represents a number in the form of p/q, where p and q are integers, and q ≠ 0.

For example:

-> The rational number 0.75 can be written in standard form as 3/4
-> The rational number -1.5 can be written in standard form as -3/2
-> The rational number 7 can be written in standard form as 7/1.

Rational Numbers on the Number Line

A number line is a visual representation of numbers arranged in a linear order. The numbers are placed along a line so that the distance between two numbers represents the magnitude of the difference between them.

On the number line, rational numbers are the numbers that can be expressed as a fraction of two integers. This includes both positive and negative numbers, as well as decimals that can be expressed as a fraction.

Example:

To put 2/5 on the number line, follow these steps:

1. Divide the number line into equal parts: Start by dividing the number line into 10 equal parts, each representing 1/5. You can use a ruler or a straight edge to draw the divisions.

2. Locate 2/5: To locate 2/5, count four parts from 0 on the number line. You will find 2/5.

3. Place a circle: Place a circle on the second part of the number line to represent 2/5.

The number line needs to be marked with a circle at the position corresponding to 2/5, indicating its placement on the number line.

Rational Numbers between Two Rational Numbers

Between any two rational numbers, there are infinitely many other rational numbers. This is because a rational number is defined as a number that can be expressed as a ratio of two integers. Therefore, as long as the two integers in the ratio are different, there will always be a different rational number.

For example, between the numbers 1/2 and 3/4, there are an infinite number of rational numbers such as 2/3, 5/7, 17/23, etc.

Properties of Rational Numbers

In this section, let’s learn the Properties of rational numbers with examples, Rational numbers are numbers that can be expressed as a fraction of two integers.

The following are the properties of rational numbers:

Closure Property of Rational Numbers

The result of arithmetic operations such as addition, subtraction and multiplication performed on two rational numbers will always be another rational number. For any rational number a/b and c/d, the results of addition, subtraction and multiplication are rational numbers.

Addition: a/b + c/d = Rational number
Subtraction: a/b – c/d = Rational number
Multiplication: a/b x c/d = Rational number

Example: Consider the two rational numbers 2/3 and 1/2. The result of their addition is (2/3) + (1/2) = 7/6.

Since 7/6 can be written in the form of p/q where p and q are integers and q ≠ 0, it is a rational number and therefore satisfies the closure property of rational numbers.

Commutative Property of Rational Numbers

The order in which rational numbers are added or multiplied does not affect the result.
a/b + c/d = c/d + a/b

For example, (2/3) + (3/4) = (3/4) + (2/3)

Associative Property of Rational Numbers

The grouping of rational numbers in addition or multiplication does not affect the result. If a/b, c/d and e/f are rational numbers where b, d and f are non-zero.

Therefore, (a/b + c/d) + e/f = a/b + (c/d + e/f)

For example, (2/3 + 4/5) + (3/4) = 2/3 + (4/5 + 3/4)

Distributive Property of Rational Numbers

The product of a rational number and the sum or difference of two rational numbers is equal to the sum or difference of the product of the rational number and each of the other two rational numbers.

For example, 2/3 (4/5 + 3/4) = 2/3 (4/5) + 2/3 (3/4)

Identity Property of Rational Numbers

Existence of Additive Identity: The additive identity of rational numbers is 0. This means that for any rational number “a”, there exists an additive identity “0” such that a + 0 = a. This property is true for all rational numbers.

For example, -5 + 0 = -5.

Existence of Multiplicative Identity: The multiplicative identity of rational numbers is 1. This means that for any rational number “a”, there exists a multiplicative identity “1” such that a x 1 = a. This property is true for all rational numbers.

For example, 2/9 x 1 = 2/9.

Inverse Property of Rational Numbers

Existence of Additive Inverse: For any number non-zero rational number “a”, its additive inverse is “-a”, which satisfies the property that
a + (-a) = 0.

For instance, the additive inverse of 5 is -5 similarly, the additive inverse of -7/2 is 7/2.

Existence of Multiplicative Inverse: For any non-zero rational number “a”, there exists a multiplicative inverse “b” such that
a x b = 1. The multiplicative inverse is also referred to as the reciprocal of the rational number.

Consider the rational number 2/3. Its multiplicative inverse is a rational number that, when multiplied by 2/3, results in 1.
2/3 x 3/2 = 1

NOTE: The only rational number that does not have a multiplicative inverse is 0 because any number multiplied by 0 is always equal to 0, and not 1.

Chart on Properties of Rational Numbers