10th Class Mathematics Real Numbers Introduction

         Introduction

Previously we have studied about various types of numbers like natural numbers, whole numbers, integers, fractions and its decimal representation, rational numbers along with its different operations and properties. In this chapter, we will study about a new number system known as REAL NUMBER which includes rational and irrational numbers.  

            Representation of Numbers on Number Lines

It is a way to represent numbers on a line with the help of diagram.  

            Representation of Integers on Number Line  

Take a line AB extended infinitely in both direction. Take a point O on it and represent it as zero (0). Mark points on line at equal distances on both sides of O. Equal distances are taken as per our convenience and take it as a unit.

On the right of O positive integers 1,2,3,4,5 etc. are indicated at the distance of 1 unit, 2 unit, 3 unit ,4 unit ,5 unit etc. respectively.

On the left of O negative integers -1,-2,-3,-4 etc. are indicated at a distance of 1 unit, 2 unit, 3 unit, 4 unit etc. respectively

            Representation of Rational Numbers on a Number Line

Take a line extended infinitely in both direction. Take a point O on it and represent it as zero (0). Mark points on line at equal distances on both sides of O. Equal distances are taken as per our convenience and take it as a unit. Suppose OP=1 unit.

Let the midpoint of OP is Q, .Therefore, \[OQ=\frac{1}{2}\] unit. Now we mark different points on the number line on right as well as left of O taking OQ =1/2 units.

The above given number line shows different rational numbers with denominator as 2.  

Important Points Related to Number Line which Represents Rational Number

(i) A particular point on the number line represents a particular rational number.

(ii) A rational number cannot be represented by two or more than two distinct points on a number line.

(iii) There are infinite points between two distinct points on a line, hence, there are infinite rational numbers between two rational numbers.  

Method to find Rational Numbers between Two Rational Numbers

(i) Suppose A and B be two rational number then a rational number which is in between A and B is \[\frac{1}{2}(A+B)\].

(ii) Suppose A and B be two rational number in which A < B, then the n rational numbers between A and B are \[(A+x),(A+2x),\] .............. \[(A+nx)\].  Where \[x=\frac{B-A}{n+1}\]