Which of the following are true statements? i.) Addition is associative for rational numbers. ii.) Subtraction is associative for rational numbers. iii.)Multiplication is associative for rational numbers. iv.) Division is associative for rational numbers.
Answers
Answer:
There are some properties of rational numbers like closure property, commutative property and associative property.
Let us explore these properties on the four binary operations (Addition, subtraction, multiplication and division) in mathematics.
Addition
(i) Closure Property :
The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q is closed under addition
If a/b and c/d are any two rational numbers, then
(a/b) + (c/d) is also a rational number
Example :
2/9 + 4/9 = 6/9 = 2/3 is a rational number
(ii) Commutative Property :
Addition of two rational numbers is commutative.
If a/b and c/d are any two rational numbers, then
(a/b) + (c/d) = (c/d) + (a/b)
Example :
2/9 + 4/9 = 6/9 = 2/3
4/9 + 2/9 = 6/9 = 2/3
So,
2/9 + 4/9 = 4/9 + 2/9
(iii) Associative Property :
Addition of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b + (c/d + e/f) = (a/b + c/d) + e/f
Example :
2/9 + (4/9 + 1/9) = 2/9 + 5/9 = 7/9
(2/9 + 4/9) + 1/9 = 6/9 + 1/9 = 7/9
So,
2/9 + (4/9 + 1/9) = (2/9 + 4/9) + 1/9
(iv) Additive Identity :
The sum of any rational number and zero is the rational number itself.
If a/b is any rational number, then
a/b + 0 = 0 + a/b = a/b
Zero is the additive identity for rational numbers.
Example :
2/7 + 0 = 0 + 2/7 = 2/7