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Commutative Property

Rational Numbers II - Concepts

Identity Property

Commutative Property

Multiplicative Inverse

Division of Rational Numbers

Multiplication of Rational Numbers

Rational Numbers between Two Rational Numbers

Properties of Rational Numbers

Simplification of Rational Numbers

Class - Navodaya Entrance 9th Subjects

Concept Explanation

Commutative Property

Commutative property of addition in rational numbers:

The addition of rational numbers is commutative i.e. if $large frac{a}{b};and;frac{c}{d}$ are any two ratoinal numbers, then

$large frac{a}{b}+frac{c}{d}=frac{c}{d}+frac{a}{b}$

Verification: In order to verify this property, let us consider two expressions

$large frac{5}{6}+frac{-4}{9}=frac{-4}{9}+frac{5}{6}$

We have,

$large frac{-4}{9}+frac{5}{6}=frac{-8+15}{18}=frac{7}{18}$ and, $large frac{5}{6}+frac{-4}{9}=frac{15}{18}+frac{-8}{18}=frac{15+(-8)}{18}=frac{7}{18}$

$large therefore ;frac{5}{6}+frac{-4}{9}=frac{-4}{9}+frac{5}{6}$

Similarly, it can be verified for others pairs of rational numbers.

Commutative property of subtraction in rational numbers:

The subtraction of rational numbers is not always commutative. That is, for any two rational numbers $large frac{a}{b}$ and $large frac{c}{d}$ , we have

$large frac{a}{b}-frac{c}{d}neq frac{c}{d}-frac{a}{b}$

For example, $large frac{2}{3}-frac{1}{6}=frac{2}{3}+frac{-1}{6}=frac{2times 2+(-1)times 1}{6}=frac{3}{6}=frac{1}{2}$

and, $large frac{1}{6}-frac{2}{3}=frac{1}{6}+frac{-2}{3}=frac{1+(-2)times 2}{6}=frac{1-4}{6}=frac{-3}{6}=frac{-1}{2}$

$large therefore ;frac{2}{3}-frac{1}{6}neq frac{1}{6}-frac{2}{3}$

Commutative property of multiplication in rational numbers:

The multiplication of rational numbers is commutative. That is, if $large frac{a}{b}$ and $large frac{c}{d}$ are any two raitonal numbers, then

$large frac{a}{b}times frac{c}{d}=frac{c}{d}times frac{a}{b}$

Verification: We have,

(i) $large frac{3}{4}times frac{5}{7}=frac{3times 5}{4times 7}=frac{15}{28}$ and, $large frac{5}{7}times frac{3}{4}=frac{5times 3}{7times 4}=frac{15}{28}$

$large therefore ;frac{3}{4}times frac{5}{7}=frac{5}{7}times frac{3}{4}$

(ii) $large frac{-5}{12}times frac{-3}{4}=frac{-5times -3}{12times 4}=frac{15}{48}=frac{5}{16}$ and, $large frac{-3}{4}times frac{-5}{12}=frac{-3times -5}{4times 12}=frac{15}{48}=frac{5}{16}$

$large therefore ;frac{-5}{12}times frac{-3}{4}=frac{-3}{4}times frac{-5}{12}$

Commutative property of division in rational numbers:

For any rational number $large frac{a}{b}$ , we have

$large frac{a}{b}div 1=frac{a}{b}$ and $large frac{a}{b}div (-1)=-frac{a}{b}=frac{-a}{b}$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Name the property of multiplication illustrated by the following: -11/13 × -17/5 = -17/5 × -11/13
A. Distributive property	B. Associative property	C. Commutative property	D. Closure property
Right Option : C
View Explanation

Question : 2

Name the property of rational numbers illustrated by : (2/9 + 4/9) = (4/9 + 2/9)
A. Distributive property	B. Associative property	C. Commutative property	D. Closure property
Right Option : C
View Explanation

Question : 3

7a + 2 = a(7) + 2, is explained by which property?
A. Associative over addition	B. Associative over multiplication	C. Commutative over multiplication	D. Commutative over Addition
Right Option : C
View Explanation