Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.
Rational numbers are closed under addition:-
Example:- 5/4 + 1/2
The LCM of the denominators 4 and 2 is 4
∴ (5/4) = [(5×1)/ (4×1)] = (5/4)
and (1/2) = [(1×2)/ (4×2)] = (1/4)
Then,
= 5/4 + ¼
= (5 + 1)/ 4
= 6/4
= 3/2 is a rational number
Rational numbers are closed under subtraction:-
Example:- 5/4 – 1/2
The LCM of the denominators 4 and 2 is 4
∴ (5/4) = [(5×1)/ (4×1)] = (5/4)
and (1/2) = [(1×2)/ (4×2)] = (1/4)
Then,
= 5/4 – ¼
= (5 – 1)/ 4
= 4/4
= 1 is a rational number
Rational numbers are closed under addition:-
Example:- 5/4 × 1/2
= 5/8 is a rational number.
For any rational number x, x ÷ 0 is not defined,
Hence not all rational numbers are closed under division. We can say that except zero, all rational numbers are closed under division.
Example, ¾ ÷ 4/5
= ¾ × 5/4
= 15/16 is a rational number.