Question

1. Verify the closure property of rational numbers (addition, subtraction, multiplication and division) for −9/ 11 and 3/ 5

Answer 1

Answer:

Here, a = $\frac{-9}{11}$ and b = $\frac{3}{5}$

A.   If a and b are two rational numbers and their sum  c = a + b  is  also a rational number, then the two rational numbers a and b is said to satisfy Closure-Property of Addition.

a + b = $\frac{-9}{11} + \frac{3}{5}$ = $\frac{-45 + 15}{55}$ =  $\frac{-30}{55}$ which is also a rational number.

This means that the rational numbers are closed under addition.

B.  If a and b are two rational numbers and their difference  c = a - b  is  also a rational number, then the two rational numbers a and b is said to satisfy Closure-Property of Subtraction.

a - b = $\frac{-9}{11} - \frac{3}{5}$ = $\frac{-45 - 15}{55}$ =  $\frac{-60}{55}$ which is also a rational number.

This means that the rational numbers are closed under subtraction.

C. If a and b are two rational numbers and their product  c = a x b  is  also a rational number, then the two rational numbers a and b is said to satisfy Closure-Property of Multiplication.

a x b = $\frac{-9}{11}$ x $\frac{3}{5}$ = $\frac{-27}{55}$   which is also a rational number.

This means that the rational numbers are closed under multiplication.

D. If a and b are two rational numbers and the quotient  c = a ÷ b  is  also a rational number, then the two rational numbers a and b is said to satisfy Closure-Property of Division.

a ÷ b = $\frac{-9}{11}$ ÷ $\frac{3}{5}$  =  $\frac{-9}{11}$ x $\frac{5}{3}$= $\frac{-15}{11}$   which is also a rational number.

This means that the rational numbers are closed under division for the given set of numbers.

But we know that any rational number n, n ÷ 0 is not defined. So rational numbers are not closed under division.