Reciprocal of a Rational Number
We will learn the reciprocal of a rational number.
For every non-zero rational number a/b there exists a rational number b/a such that
a/b × b/a = 1 = b/a × a/b
The rational number b/a is called the multiplicative inverse or reciprocal of a/b and is denoted by (a/b)-1.
The reciprocal of 12 is 1/12
The reciprocal of 5/16 is 16/5.
The reciprocal of 3/4 is 4/3 i.e., (3/4)^-1 = 4/3.
The reciprocal of -5/12 is 12/-5 i.e., (-5/12)^-1 = 12/-5.
The reciprocal of (-14)/17 is 17/-14 i.e., (-17)/14.
The reciprocal of -8 is 1/-8 i.e., (-1)/8.
The reciprocal of -5 is 1/-5, since -5 × 1/-5 = -5/1 × 1/-5 = -5 × 1/-5 × 1 = 1.
Note: The reciprocal of 1 is 1 and the reciprocal of -1 is -1. 1
and -1 are the only rational numbers which are their own reciprocals. No other
rational number is its own reciprocal.
We know that there is no rational number which when multiplied with 0, gives 1. Therefore, rational number 0 has no reciprocal or multiplicative inverse.
Solved example on reciprocal of a rational number:
1. Write the reciprocal of each of the following rational numbers:
(i) 5
(ii) -15
(iii) 7/8
(iv) -9/13
(v) 11/-19
Solution:
(i) The reciprocal of 5 is 1/5 i.e., (5)^-1 = 1/5.
(ii) The reciprocal of -15 is 1/-15 i.e., (-15)^-1 = 1/-15.
(iii) The reciprocal of 7/8 is 8/7 i.e., (7/8)^-1 = 8/7.
(iv) The reciprocal of -9/13 is 13/-9 i.e., (-9/13)^-1 = 13/-9.
(v) The reciprocal of 11/-19 is -19/11 i.e., (11/-19)^-1 = -19/11.
2. Find the reciprocal of 3/7 × 2/11.
Solution:
3/7 × 2/11
= (3 × 2)/(7 × 11)
= 6/77
Therefore, the reciprocal of 3/7 × 2/11 = Reciprocal of 6/77 = 77/6.
3. Find the reciprocal of -4/5 × 6/-7.
Solution:
-4/5 × 6/-7
= (-4 × 6)/(5 × -7)
= -24/-35
= 24/35
Therefore, the reciprocal of -4/5 × 6/-7 = Reciprocal of 24/35 = 35/24.
● Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
Standard form of a Rational Number
Equality of Rational Numbers using Standard Form
Equality of Rational Numbers with Common Denominator
Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
Rational Numbers in Descending Order
Representation of Rational Numbers on the Number Line
Rational Numbers on the Number Line
Addition of Rational Number with Same Denominator
Addition of Rational Number with Different Denominator
Properties of Addition of Rational Numbers
Subtraction of Rational Number with Same Denominator
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Rational Expressions Involving Addition and Subtraction
Simplify Rational Expressions Involving the Sum or Difference
Multiplication of Rational Numbers
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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