Rational Numbers

Types of Numbers

Natural Numbers

  • Set of counting numbers is called the Natural Numbers.

Whole numbers

  • Set of Natural numbers plus Zero is called the Whole Numbers

Note:
All natural Number are whole number but all whole numbers are not natural numbers

Examples:

2 is Natural Number
-2 is not a Natural number
0 is a Whole number

Integers

In Number System, Integers is the set of all the whole number plus the negative of Natural Numbers

Note

  • Integers contains all the whole number plus negative of all the natural numbers.
  • The natural numbers without zero are commonly referred to as positive integers.
  • The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer.
  • Natural numbers with zero are referred to as non-negative integers.
  • The natural numbers form a subset of the integers.

What is a Rational Number ?

  • A number is called rational if we can write the number in the form of p/q where p and q are integers and q ≠ 0.
    i.e.,
  • 1 = 1/1
  • 2 = 2/1
  • 0 = 0/1

and 

  • 5/7
  • (-30)/17
  • 6/(-11)
  • (-4)/(-5)

are all rational numbers.

Properties on Rational Numbers

(i) Closure Property

Rational numbers are closed under addition, subtraction and multiplication.

Division :

(-3)/5 ÷ 2/3 = (-9)/10, which is also a rational number.

but, For any rational number a, a ÷ 0 is not defined. So, rational number are not closed under division.

However, if we exclude zero then the rational numbers are closed under
division.

(ii) Commutativity

Addition:

  • Two rational numbers can be added in any order, i.e.,
    commutativity holds for rational numbers under addition.
  • For any two rational number a and b, a + b = b + a.

Subtraction:

  • For any two rational number a and b, a - b ≠ b - a.
  • Subtraction is not associative for rational numbers.

Multiplication:

  • Multiplication is commutative for rational
    numbers.
  • In general, a × b = b × a, for any two rational numbers a and b.

Division:

  • In general, a ÷ b ≠ b ÷ a , for any two rational numbers a and b.
  • Hence, division is not Cumulative for rational numbers.

(iii) Associativity

Addition:

  • Addition is associative for rational numbers,
  • For any three rational numbers a, b and c, a + (b + c) = (a + b) + c.

Subtraction:

  • Subtraction is not associative for rational numbers.

Multiplication:

  • Multiplication is associative for rational number
  • For any three rational numbers a, b and c, a × (b × c) = (a × b) × c.

Division:

  • Division is not associative for rational numbers.

(iv)Distributivity of multiplication over addition for rational number

  • For all rational numbers a, b and c, a(b + c) = ab + ac

(v)Distributivity of multiplication over subtraction for rational number:

  • For any three rational numbers a, b and c, a (b – c) = ab – ac

Additive Identity/Role of Zero

We observe
a + 0 = 0 + a = a, where a is a whole number
b + 0 = 0 + b = b, where b is an integer
c + 0 = 0 + c = c, where c is a rational number

Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.

Multiplicative identity/Role of one

We observe
a × 1 =  = a, where a is a whole number
b × 1 = b, where b is an integer
c × 1 = c, where c is a rational number

1 is the multiplicative identity for rational numbers. It is the multiplicative identity for integers and whole numbers as well

Additive Inverse

We observe
a + (-a) = 0, where a is a whole number
b +(-b)  = 0, where b is an integer
(a/b) + (-a/b) = 0, where a/b is a rational number

So we say that (-a/b) is the additive inverse of a/b  and a/b is the additive inverse for (-a/b)

Example

What is the additive inverse of following numbers
a) 1/6
b) -8/9

Solution:
Additive Inverse of any rational number a/b is defined as -a/b  so that (a/b) +(-a/b)=0
So   for 1/6, additive inverse will be -1/6
Similarly for -8/9 ,it will be  8/9

Reciprocal or multiplicative inverse

The multiplicative inverse of any rational number a/b is defined as b/a
  so that (a/b) x (b/a) =1

Zero does not have any reciprocal or multiplicative inverse

Example
 Find the multiplicative inverse of the following.
-54
-11/12

Solution

The multiplicative inverse of any rational number a/b is defined as b/a  so that (a/b) x (b/a) =1
So answer would be
-54=-54/1 multiplicative inverse will 1/-54 =-1/54
Similarly for -11/12, the multiplicative inverse will be -12/11

Representation of Rational Number on a number line

  • We draw a line.
  • We mark a point O on it and name it 0. Mark a point to the right of
    0. Name it 1. The distance between these two points is called unit
    distance.
  • Mark a point to the right of 1 at unit distance and name it 2.
  • Proceeding in this manner, we can mark points 3, 4, 5,
  • Similarly we can mark – 1, – 2, – 3, – 4, – 5, ……… to the left of 0.
    This line is called the number line.
  • This line extends indefinitely on both sides.
If the Rational number is in the form of Numerator and denominator then to represent it on the number line follow the steps mentioned in the video.

 

    Note :

    • The positive rational numbers are represented by points on the number
      line to the right of O .
    • the negative rational numbers are represented by points on the number line to the left of O.
    • Any rational number can be represented on this line. The denominator of the rational number indicates the number of equal parts into which the first unit has been divided whereas the numerator indicates as to how many of these parts are to be taken into consideration.

    Rational Numbers Between Two Rational Numbers

    • We can find infinitely many rational numbers between any two given
      rational numbers.
    • We can take the help of the idea of the mean for this
      purpose.
    • Between two rational numbers x and y, there exists a rational number = (x+y)/2
    Alternative Method to find Rational numbers between two Rational Numbers

    Make the denominator Same and find Rational numbers accordingly.