ICSE Class 8 Maths Number Systems Chapter 04 Real Numbers

Unit Two: Numbers

Real Numbers

Directed Numbers

HCF and LCM

Fractions and Decimals

Approximation

Squares and Square Roots

Powers and Roots

Let's Recap

Problem 1: Write the following numbers in ascending and descending order

(i) \(\frac{7}{8}, \frac{3}{10}, \frac{2}{7}, \frac{4}{5}, \frac{11}{24}, 1\frac{1}{3}\)

(ii) 1.7117, 7.1771, 1.7711, 7.7171, 1.7771

(iii) 6.213, 6.231, 6.132, 6.321, 6.3

(iv) 9, 9³, \(\sqrt[3]{8}\), 8³, \(\sqrt{9}\), 8

Problem 2

The additive inverse of -24 is multiplied with the multiplicative inverse of \(2\frac{2}{3}\). What is the product obtained?

Problem 3

By how much does \(-4\frac{1}{6}\) have to be increased in order to get \(-2\frac{5}{6}\)?

Problem 4

Mango trees cover 0.35 portion of an orchard, while guava trees grow on 0.26 portion. If the rest of the orchard has 3042 litchi trees, how many mango trees are there in the orchard?

Hmm... 120 square flowerbeds in a square field... We have made a mistake here. There cannot be 120 square flowerbeds in a row.

Collating data on their projects to create a website...

Well, why don't we change the number to a perfect square, say 121 or 144?

SILENCE PLEASE

4. Real Numbers

Natural Numbers

Whole Numbers

Integers

Fractions

Rational Numbers

Irrational Numbers

Operations Involving Irrational Numbers

Properties of Irrational Numbers

Rationalising Factor

Real Numbers

Properties of Real Numbers

Introduction

We have learnt about natural numbers, whole numbers, integers, and rational numbers in previous classes. In this chapter, we will briefly recall what was learnt earlier and extend the system of these numbers to the set of real numbers.

Whole numbers may be represented on a number ray with 0 as its end point and all subsequent numbers on its right at an equal distance from each other.

0 1 2 3 4 5 6 7

Natural Numbers

The set of natural numbers is the infinite set of counting numbers beginning with 1.

\[N = \{1, 2, 3, 4, 5, 6, \ldots\}\]

Natural numbers may be represented on a number ray with 1 as its end point and all subsequent numbers on its right at an equal distance from each other.

1 2 3 4 5 6 7 8

Whole Numbers

The set of natural numbers along with the digit 0 form the infinite set of whole numbers.

\[W = \{0, 1, 2, 3, 4, 5, 6, \ldots\}\]

Integers

The additive inverse of a natural number is a negative integer. The sum of a number and its additive inverse is 0. The set of whole numbers, along with the set of the additive inverse of all natural numbers, forms the infinite set of integers.

Integers are represented by I or Z.

\[Z = \{\ldots, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, \ldots\}\]

Integers may be represented on a number line with 0 as its mid-point, all natural numbers to its right, and all negative integers to its left.

-5 -4 -3 -2 -1 0 1 2 3 4 5

Natural numbers are positive integers. 0 is an integer which is neither positive nor negative.

Fractions

Fractions are numbers that are written in the form \(\frac{a}{b}\), where a and b are natural numbers.

Fractions are represented on a number ray to the right of zero.

0 \(\frac{1}{4}\) \(\frac{1}{2}\) \(\frac{3}{4}\) \(\frac{4}{4}\) \(\frac{5}{4}\) \(\frac{3}{2}\) \(\frac{7}{4}\)

Divide the distance between 0 and -1 into seven equal parts. The distance to the left of 0, till the sixth of these parts, represents \(-\frac{6}{7}\). Now divide the distance between 1 and 2 into eight equal parts. The distance to the right of 1, till three of these parts, represents \(+1\frac{3}{8}\).

-2 \(-\frac{6}{7}\) 0 \(+1\frac{3}{8}\) +2

Rational Numbers

All numbers that can be written in the form \(\frac{p}{q}\), where p and q are integers, but q ≠ 0, form the set of rational numbers. It is represented by Q.

\[Q = \{\ldots, -\frac{3}{4}, \ldots, -\frac{1}{7}, \ldots, +2.4, \ldots, +5\frac{1}{5}, \ldots\}\]

Thus, rational numbers include:

1. \(\frac{p}{q} = \frac{8}{2} = 4\) (natural numbers)

2. \(\frac{p}{q} = \frac{0}{7} = 0\) (whole numbers)

3. \(\frac{p}{q} = \frac{-14}{7} = -2\) (integers)

4. \(\frac{p}{q} = \frac{5}{7}\) (fractions)

5. \(\frac{p}{q} = \frac{-8}{9}\) (negative fractions)

Representation of a Rational Number on the Number Line

Rational numbers may be represented on the number line with 0 as its mid-point.

Example 1: Represent \(-\frac{6}{7}\) and \(\frac{3}{8}\) on the number line.

If -6 is divided by 7, it is apparent that the quotient will be less than 0 but more than -1. Thus \(-\frac{6}{7}\) will lie between -1 and 0 on the number line. Similarly, \(\frac{3}{8}\) will lie between +1 and +2 on the number line.

Insertion of Rational Numbers

Rational numbers are very densely packed on the number line.

There can be infinite rational numbers between two given rational numbers. This is why there can be no predecessor or successor of a rational number.

The common fraction, also known as vulgar fraction, obtained by adding the numerators and denominators of any two given common fractions, will always lie between the two on the number line.

Example 2: Find two rational numbers between -8.17 and -8.18.

The average of the given numbers is given by

\[\frac{-8.17 + (-8.18)}{2} = \frac{-16.35}{2} = -8.175\]

The average of -8.17 and -8.175 = \(\frac{-8.17 + (-8.175)}{2}\)

\[= \frac{-16.345}{2} = -8.1725\]

Thus, 2 rational numbers between -8.17 and -8.18 are -8.175 and -8.1725.

Example 3: Find two rational numbers between \(\frac{4}{7}\) and \(\frac{5}{7}\).

The given fractions are \(\frac{4}{7}\) and \(\frac{5}{7}\).

Now, \(\frac{4 + 5}{7 + 7} = \frac{9}{14}\) will lie between \(\frac{4}{7}\) and \(\frac{5}{7}\).

\(\frac{4 + 9}{7 + 14} = \frac{13}{21}\) will lie between \(\frac{4}{7}\) and \(\frac{9}{14}\).

Thus, 2 rational numbers between \(\frac{4}{7}\) and \(\frac{5}{7}\) are \(\frac{9}{14}\) and \(\frac{13}{21}\).

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