# Chapter 9 – Rational Numbers

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

In the previous chapter, we learned about the number system, and in this chapter, we will learn about rational numbers.

Let us quickly revise what we have learnt so far about the number system. In the number system, natural numbers are counted as 1,2,3 and so on. And when 0 is added to the natural numbers, it becomes whole numbers.

Integers are negatives of natural numbers added to the number system.

#### What Are Rational Numbers?

Any number that can be expressed in the form of p/q, where p and q are integers and q 0, is a rational number.

Thus, rational numbers are a mathematical expression with numerator and denominator as integers, and where the denominator is not a zero. The division of any rational number will always be either a terminating or repeating decimal.

A whole number is always a rational number.

#### Positive and Negative Rational Numbers

When p and q are integers, a rational number is in the form of p/q and q has to be a non-zero number. Rational numbers can be both positive and negative.

The point that differentiates both positive and negative rational numbers is that, in positive rational numbers, both numerators and denominators are of the same sign. For example, 8/9, 5/7,2/6, etc.

In negative rational numbers, the numerator and the denominator are of opposite signs. For example, -6/12, 33/-100, 45/-125, etc.

#### Rational Numbers on a Number Line

Just like whole numbers and integers, rational numbers can also be represented on a number line. The number zero is the origin of any number line, and all positive numbers are represented on the right-hand side of zero, and all negative numbers are represented in the left-hand side.

In terms of rational numbers represented on a number line, there are two types of fractions— proper fraction and improper fraction.

In a proper fraction, the value of the numerator is always less as compared to the denominator. Thus the rational number can be easily represented on a number line, as in proper fraction its value is less than 1 and more than 0.

In an improper fraction, the denominator’s value is less compared to the numerator, so the value of a given rational number is greater than 1. This calls for converting the improper fraction into a mixed fraction, and this process will help you locate the position of rational numbers on the number line.

#### Rational Numbers in Standard Form

When the numerator and denominator have no common factor other than 1 and the denominator of the rational number is a positive integer, the rational number is said to be in standard form.

#### Comparison of Rational Numbers

We compare two negative rational numbers by ignoring their negative signs and reversing its order. Similarly, when comparing negative and positive rational numbers, a positive rational number will always be more than the negative rational numbers.

#### Operations on Rational Numbers

In the multiplication of any rational number, the multiplicative inverse and the reciprocal of any given fraction is the same because a rational number is denoted by p/q.

Here is an example of multiplication of a rational number
7/8 is a rational number, and the multiplicative inverse of it is 8/7, such that (7/8)x(8/7) = 1.

If a and b are two rational numbers divided, then the result of the division of these two rational numbers where b0 can be obtained by multiplication of a to reciprocal of b.

Eg. a/b is to be divided by z/q, then:

(a/b) (z/q) = aq/bz

For the addition of any two rational numbers (i.e. fraction), the denominators should be the same. For example, if a/b is to be added to z/q, then a common denominator has to be derived, such as:

(a/b) + (z/q) = (a/q+bz)/bq

For subtraction of a rational number too, the denominators have to be the same.

(a/b) – (z/q) = (a/q-bz)/bq.

#### Properties of Rational Numbers

• If a rational number’s denominator is a positive integer and the numerator and denominator have no common factor, then this rational number is called “the standard form’.
• When zero is added to a rational number, it will give the same number.
• When a rational number is divided, added or multiplied, the results will always be the same.
• If the numerator and the denominator are either multiplied or divided by the same factor, the rational number will remain the same.
• Rational numbers are always closed under subtraction, addition and multiplication. Note: Division is not defined by zero, so it is not under closure property.
• The denominator in a rational number cannot be zero.
• The denominator and numerator in a rational number have to be integers.

#### Conclusion

Thus we now know that rational numbers are sets of fractions, integers and decimal numbers represented in the form of a/b, where a and b are integers and b 0. The set of rational numbers will always be smaller compared to irrational numbers.

#### FAQs

1. What are five examples of rational numbers?
Any number that can be expressed as the quotient of integers is a rational number.Some examples of rational numbers are:
-55/321, 3/122, 66/-789, 54/55, 66/111
2. Is seven a rational number?
The digit seven (7) is definitely a rational number because it can be expressed as a quotient of two integers, 7/1.
3. Is the number 0 rational number?
The number 0 is an example of a rational number since rational numbers include positive, negative and zero.
4. Is five a rational number?
Five can be written as a fraction, 5/1; therefore, it is definitely a rational number. Any number that can be written as a fraction is a rational number.
5. How can you identify a rational number?
a) Any number that can be written in fraction form is a rational number.
b) A fraction with a non-zero denominator is a rational number.
c) A number that can be expressed in a ratio form is a rational number.
d) Rational numbers include all integers and decimals.

We hope the concept of rational numbers is clear to you. If you have any doubts or queries related to rational numbers, download the MSVgo app now. It has an extensive video library explaining topics with examples.

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