Question

Which of the following sets are closed under division? Select all that apply.

integers
irrational numbers
whole numbers
polynomials
none of the above

Answer 1

The appropriate choice is probably
  none of the above


While the inverse of any irrational number is irrational, their ratio my not be, for example (√8)/(√2) is rational.

Answer 2

Answer:

None of the Above

Step-by-step explanation:

Integers:

Take $a=1,b=2$

Here, $\frac{a}{b}=\frac{1}{2}$

Since $\frac{1}{2}$ is not an integer , integers are not closed under division .

Irrational Numbers:

Take $a=\sqrt{3},b=\sqrt{27}$

Here,$\frac{a}{b}=\frac{\sqrt{3} }{\sqrt{27} }=\frac{1}{\sqrt{9} }=\frac{1}{3}$

Since $\frac{1}{3}$ is not an irrational number , irrational numbers are not closed under division.

Whole Numbers:

Take $a=2,b=3$

Here, $\frac{a}{b} =\frac{2}{3}$

Since $\frac{2}{3}$ is not a whole number, whole numbers are not closed under division.

Polynomials:

Take two polynomials as $2x+3 , 0$

Here, $\frac{2x+3}{0}$ is not defined.

Since $\frac{2x+3}{0}$ is not a polynomial, polynomials are not closed under division .

So, the correct option is none of the above