TYPES OF NUMBERS AND NUMBER LINE

Published on: March 22, 2024

Introduction

A number system is a way to express the numbers of a given set using digits or symbols in a consistent manner. Let us recall some of them that we studied earlier.

 

Natural Numbers

The numbers 1, 2, 3, 4,... are called natural numbers. They are usually represented by the symbol N.

 

Whole Numbers 

The numbers 0, 1, 2, 3, 4,... are called whole numbers. They are usually represented by the symbol W.

 

Integers 

The numbers −3,−2,−1, 0, 1, 2, 3,... are called integers. They are usually represented by the symbol Z, which is the abbreviation of the German word 'Zahlen', which means ‘to count’.

Co-prime integers: Two integers are said to be co-prime if they have no common factors except 1.

 

Rational Numbers

A rational number is a number of the form pq, where and are integers and q ≠ 0. The word 'rational' is derived from the word 'ratio'. The set of all rational numbers is represented by the symbol Q, which comes from the word 'Quotient'.

-2, 0, 3,  23, 94,-23 etc. are rational numbers. In other words, all the integers, whole numbers, and natural numbers are included in the collection of rational numbers.

 

Notes:

* Between any two given rational numbers, there are infinitely many rational numbers.

* Equivalent rational numbers: Equivalent rational numbers are obtained by either multiplying or dividing both the numerator and the denominator by the same value.

 

For example, to find the equivalent rational numbers of the number 35, multiply both the numerator and the denominator by a number, say 2.

612,1220, 2440 etc. are rational numbers equivalent to the number 35.

 

* Finding a rational number between two rational numbers:

 

1. If ab and cd are two rational numbers, then a+cb+d is a rational number that lies between ab and cd

 

2. If ab and cb are two rational numbers, then a+c2b is a rational number that lies between ab and cb.

 

Irrational Numbers

Numbers that are not rational are called irrational numbers, i.e., numbers that cannot be written as simple fractions.

2, 2.4747..,π etc, are irrational numbers. 

If a and b are two distinct positive irrational numbers, then ab is an irrational number lying between a and b. 

 

Real Numbers

Every real number is either a rational number or an irrational number.

 

Q1: Find a rational number between 12 and 13

 

Solution : Add the numerators: 1 + 1 = 2

Add the denominators: 2 + 3 = 5

25 is a rational number lies between 12 and 13

 

Alternative method: Make the denominators of both the fractions common

12=1×32×3=36 

 

13=1×23×2=26

 

It is difficult to find rational numbers between 26 and 36; hence, multiply the numerator and denominator by 10.

26=2060

 

36=3060

 

Hence, the rational numbers between 2060 and 3060 are 2160,2260,..........2960.

 

∴ 2460=25  is a rational number between 12 and 13.

 

Q2: Find a rational number between 35 and 78.

 

Solution: Add the numerators: 3 + 7 = 10

Add the denominators: 5 + 8 = 13

 

1013 is a rational number between 35 and 78.

 

Note:

There are infinitely many rational numbers between any two rational numbers. We can split two numbers into smaller parts in order to find the numbers between them. It is not possible to pick up each and every number in the number line because of the infinite numbers present in it.

 

Irrational Numbers

A number that cannot be expressed in the form pqwhere 'p' and 'q' are integers and q ≠ 0 is called an irrational number. The Pythagoreans (followers of the famous mathematician Pythagoras) proved that 2 is irrational. In 425 BC, Theodorus of Cyrene demonstrated that 3, 5, 6, 7, 10, 11, 12, 13, 14, 15 & 17 are also irrationals. 

In the 1700s, Lamberte and Legendre proved that π is irrational. 

 

Examples: 

(a). 2, 3, 5, 7, 11.......

(b). 1.101001000100001......

(c). π = 3.14159...

 

The set of all rational and irrational numbers together forms the collection of real numbers, denoted by R. So, every real number is represented by a point on the number line. Also, every point on the number line represents a unique real number. This had been proved by two German mathematicians, G. Cantor and R. Dedekind. Hence, the number line is called a real number line.

 

Q1:  Locate 2 on the number line.

 

Here pythagoras theorem is taken for locating 2.

By pythagoras theorem, 

Hypotenuse= Base2 + Perpendicular2

 

22=12+12

Transfer the figure onto the number line such that the vertex C coincides with 0 on the number line.

 

Using a compass, with C as the center and CA as the radius, draw an arc touching the number line. The point P where the arc touches the number line denotes 2 on the line.

 

Q2: Locate 5 on the number line. 

 

Pythagoras theorem is taken for locating 5.

By Pythagoras theorem,

Hypotenuse= Base2 + Perpendicular2

 

52=22+12

 

Transfer the figure onto the number line such that the vertex C coincides with 0 on the number line.

 

Using a compass, with O as the center and OB as the radius, draw an arc touching the number line. The point P where the arc touches the number line denotes 5 on the line.