Chapter 9: Rational Numbers – Notes

 

Introduction to Rational Numbers

In mathematics,Rational Numbers are classified into various types, such as natural numbers, whole numbers, integers, and fractions. Chapter 9 introduces a new category called rational numbers, which expands our understanding of numbers beyond integers and fractions.

In Maths, a rational number is a type of real number, which is in the form of p/q where q is not equal to zero. Any fraction with non-zero denominators is a rational number. Some of the examples of rational numbers are 1/2, 1/5, 3/4, and so on. The number “0” is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values. Also, check irrational numbers here and compare them with rational numerals.

In this article, we will learn about what is a rational number, the properties of rational numbers along with its types, the difference between rational and irrational numbers, and solved examples. It helps to understand the concepts in a better way. Also, learn the various rational number examples and learn how to find rational numbers in a better way. To represent rational numbers on a number line, we need to simplify and write in the decimal form first.

 

What is a Rational Number?

A rational number, in Mathematics, can be defined as any number which can be represented in the form of p/q where q ≠ 0. Also, we can say that any fraction fits under the category of rational numbers, where the denominator and numerator are integers and the denominator is not equal to zero. When the rational number (i.e., fraction) is divided, the result will be in decimal form, which may be either terminating decimal or the repeating decimal.

How to identify rational numbers?

To identify if a number is rational or not, check the below conditions.

• It is represented in the form of p/q, where q≠0.
• The ratio p/q can be further simplified and represented in decimal form.

The set of rational numerals:

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

Examples of Rational Numbers: 

p	q	p/q	Rational
10	2	10/2 =5	Rational
1	1000	1/1000 = 0.001	Rational
50	10	50/10 = 5	Rational

 

Types of Rational Numbers

A number is rational if we can write it as a fraction, where both denominator and numerator are integers and the denominator is a non-zero number.

The below diagram helps us to understand more about the number sets.

 

• Real numbers (R) include all the rational numbers (Q).
• Real numbers include the integers (Z).
• Integers involve natural numbers(N).
• Every whole number is a rational number because every whole number can be expressed as a fraction

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.

Arithmetic Operations on Rational Numbers

In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how we can perform these operations on rational numbers, say p/q and s/t.

Addition: When we add p/q and s/t, we need to make the denominator the same. Hence, we get (pt+qs)/qt.

Example: 1/2 + 3/4 = (2+3)/4 = 5/4

Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.

Example: 1/2 – 3/4 = (2-3)/4 = -1/4

Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).

Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8

Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/qs

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

Multiplicative Inverse of Rational Numbers

As the rational number is represented in the form p/q, which is a fraction, then the multiplicative inverse of the rational number is the reciprocal of the given fraction.

For example, 4/7 is a rational number, then the multiplicative inverse of the rational number 4/7 is 7/4, such that (4/7)x(7/4) = 1

Rational Numbers Properties

Since a rational number is a subset of the real number, the rational number will obey all the properties of the real number system. Some of the important properties of the rational numbers are as follows:

• The results are always a rational number if we multiply, add, or subtract any two rational numbers.
• A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.
• If we add zero to a rational number then we will get the same number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

Learn more properties of rational numbers here.

Rational Numbers and Irrational Numbers

There is a difference between rational and Irrational Numbers. A fraction with non-zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by integer 2. All the numbers that are not rational are called irrational. Check the chart below, to differentiate between rational and irrational.

 

 

Rationals can be either positive, negative or zero. While specifying a negative rational number, the negative sign is either in front or with the numerator of the number, which is the standard mathematical notation. For example, we denote the negative of 5/2 as -5/2.

An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are:

Pi (π) = 3.142857…

Euler’s Number (e) = 2.7182818284590452…….

√2 = 1.414213…

How to Find the Rational Numbers between Two Rational Numbers?

There are infinite numbers of rational numbers between two rational numbers. The rational numbers between two rational numbers can be found easily using two different methods. Now, let us have a look at the two different methods.

Method 1: 

Find out the equivalent fraction for the given rational numbers and find out the rational numbers in between them. Those numbers should be the required rational numbers.

Method 2: 

Find out the mean value for the two given rational numbers. The mean value should be the required rational number. In order to find more rational numbers, repeat the same process with the old and the newly obtained rational numbers.

Summary

• Rational numbers are numbers of the form pq \frac{p}{q} qp​ (where q≠0 q \neq 0 q=0).
• They can be positive, negative, or zero and can be represented on a number line.
• Operations like addition, subtraction, multiplication, and division follow specific rules.
• Properties like closure, commutative, associative, identity, and inverse apply to rational numbers.
• Comparing and finding rational numbers between two given numbers are key skills.

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Chapter 10: Practical Geometry

 

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