Properties of Rational Numbers

Properties of Rational Numbers are the characteristics that are observed while doing mathematical operations on them. Knowing the properties of rational numbers will improve our understanding of numbers and Maths. It is must to understand properties of rational numbers to solve number related problems easily and quickly.

Properties of Rational Numbers

Just like properties of whole numbers and  integers, properties of rational numbers can be classified under Closer property, Commutative property, Associative property, Distributive property, Role of 1, Role of 0.

Properties of Rational Numbers | Closure Property

If the result obtained after performing a specific mathematical operation between any two rational numbers is always a rational number, then rational numbers are said to be closed for that operation.

Addition: Rational numbers are closed under addition.

For any two rational numbers a and b, c is also a rational number.

a + b = c , c ∈ Q

e.g.,

2/7 + 5/6 = (12+35)/42 = 47/42 (a rational number)

Subtraction: Rational numbers are closed under subtraction.

For any two rational numbers a and b, c is also a rational number.

a − b = c , c ∈ Q

e.g.,

2/7 − 5/6 = (12−35)/42 = −23/42 (a rational number)

Multiplication: Rational numbers are closed under multiplication.

For any two rational numbers a and b, c is also a rational number.

a × b = c , c ∈ Q

e.g.,

2/7 × 5/6 = 10/42 = 5/21 (a rational number)

Division: Rational numbers are not closed under division.

For any two rational numbers a and b, c may not be a rational number.

a ÷ b = c , c ∉ Q for b = 0

e.g.,

2/7 ÷ 0/6 = 2/7 × 6/0 = 12/0 (not defined, not a rational number)

Properties of Rational Numbers | Commutative Property

If the result obtained before and after changing the orders of any two rational numbers for a particular mathematical operation is same, then rational numbers are said to be commutative for that operation.

Addition: Rational numbers are commutative for addition.

for any two rational numbers a and b,

a + b = b + a

e.g.,

2/7 + 5/6 = (12+35)/42 = 47/42

5/6 + 2/7 = (35+12)/42 = 47/42

Subtraction: Rational numbers are not commutative for subtraction.

for any two rational numbers a and b,

a − b ≠ b − a

e.g.,

2/7 − 5/6 = (12−35)/42 = −23/42

5/6 − 2/7 = (35−12)/42 = 23/42 (results are not same but opposite).

Multiplication: Rrational numbers are commutative for multiplication.

for any two rational numbers a and b,

a × b = b × a

e.g.,

2/7 × 5/6 = 10/42 = 5/21
5/6 × 2/7 = 10/42 = 5/21

Division: Rational numbers are not commutative for division.

for any two rational numbers a and b,

a ÷ b ≠ b ÷ a
e.g.,
2/7 ÷ 1/3 = 2/7 × 3/1 = 6/7

1/3 ÷ 2/7 = 1/3 × 7/2 = 7/6 (results are not same but inverse).

Properties of Rational Numbers | Associative Property

If the result obtained before and after changing the grouping of rational numbers for a particular mathematical operation is same, then rational numbers are said to be commutative for that operation.

Addition: Rational numbers are associative for addition.

for any two rational numbers a and b,

(a + b) + c = a + (b + c)

e.g.,

(2/7 + 5/6) + 4/3 = 47/42 + 4/3 = (47+56)/42 = 103/42

2/7 + (5/6 + 4/3) = 2/7 + 13/6 = (12+91)/42 = 103/42

Subtraction: Rational numbers are not associative for subtraction.

for any two rational numbers a and b,

(a − b) − c ≠ a − (b − c)

e.g.,

(2/7 − 5/6) − 4/3 = −23/42 − 4/3 = (−23−56)/42 = −79/42

2/7 − (5/6 − 4/3) = 2/7 − (−3)/6 = {12−(−21)}/42 = 33/42

Multiplication: Rational numbers are associative for multiplication.

for any two rational numbers a and b,

(a × b) × c = a × (b × c)

e.g.,

(2/7 × 5/6) × 3/2 = 5/21 × 3/2 = 5/14
2/7 × (5/6 × 3/2) = 2/7 × 5/4 = 5/14

Division: Rational numbers are not associative for division.

for any two rational numbers a and b,

(a ÷ b) ÷ c ≠ a ÷( b÷ c)

e.g.,

(2/7 ÷ 1/3) ÷ 4/5 = (2/7 × 3/1) ÷ 4/5 = 6/7 × 5/4 = 15/14 ,
2/7 ÷ (1/3 ÷ 4/5) = 2/7 ÷ (1/3×5/4) = 2/7 ÷ 5/12 = 2/7 × 12/5 = 24/35

Properties of Rational Numbers | Distributive Property

Multiplying a rational number with sum of two rational numbers will give the same result if the number is first multiplied with each addend and products are then added in next step.

For all rational numbers a, b and c,

a (b + c) = ab + ac
a (b – c) = ab – ac

Properties of Rational Numbers | Role of 0

Additive Identity

Zero is called the identity for the addition of rational numbers because when we add 0 to any rational number we get the same number i.e. the number retain its identity. It is the additive identity for integers and whole numbers as well.

e.g.,

2/3 + 0 = 2/3,
−2 + 0 = −2,
21 +0 = 21

Additive Inverse (Negative) :

When we add a rational number with its negative (opposite) we get 0. Negative of a rational number is called its additive inverse.

e.g.,

Additive inverse of 2/3 is −2/3

because 2/3 + (−2/3) = 0

Additive inverse of −4/3 is 4/3

because −4/3 + 4/3 = 0

Additive inverse of any rational number 'a' \,is\,\, '-a'.

a + (-a) = 0

Properties of Rational Numbers | Role of 1

Multiplicative Identity

One is called the identity for the multiplication of rational numbers because when we multiply 1 to any rational number we get the same number i.e. the number retain its identity. It is the multiplicative identity for integers and whole numbers as well.

e.g.,

2/3 × 1 = 2/3,
−2 × 1 = −2,
21 × 1 = 21

Multiplicative Inverse (Reciprocal)

When we multiply a rational number with its reciprocal (inverse) we get 1. Reciprocal of a rational number is called its multiplicative inverse.

e.g.,

Multiplicative inverse of 2/3 is 3/2

because, 2/3 × 3/2 = 6/6 =1

Multiplicative inverse of −4/3 is −3/4

because, −2/3 × −3/4 =12/12 =1

Multiplicative inverse of any rational number 'a'\, is\, '\frac{1}{a}'.
a \times \frac{1}{a} = 1

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