Closure Property
If the result obtained after performing a specific mathematical operation between any two rational numbers is also a rational number then, the set of rational numbers is 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)
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 = −23/425/6 − 2/7 = (35−12)/42 = 23/42 (results are not same but opposite)
for any two rational numbers a and b,
a × b = b × a
e.g., 2/7 × 5/6 = 10/42 = 5/215/6 × 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 = 7/6Associative 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 = −79/422/7 − (5/6 − 4/3) = 2/7 − −3/6 = {12−(−21)}/42 = 33/42
for any two rational numbers a and b,
(a × b) × c = a × (b × c)
e.g., (2/7×5/6)×3/2 = 10/42×3/2 = 5/2/7×(5/
↬ 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 = 36/7×5/4 2 = 15/14
2/7÷(1/3÷4/5) = 2/7÷(1/3×5/4) = 2/7÷5/12 = 2/7×12/5 = 24/35Distributive Property
Multiplying a rational number with sum of two rational numbers will give same result when the number is first multiplied with each addend & then adding the products.↬ For all rational numbers a, b and c,
a (b + c) = ab + ac
a (b – c) = ab – ac
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 any 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
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 any 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 '1/a'.
a × 1/a= 1
| Rational Numbers | |