Question

Your teacher makes the following statement, “The set of whole numbers is closed under multiplication, but the set of irrational numbers is not closed under multiplication.”

What does it mean for a set of numbers to be closed under multiplication?
What example could you use to support the statement that whole numbers are closed under multiplication? Explain.
What example could you use to support the statement that irrational numbers are not closed under multiplication? Explain.

Answer 1

See explanation

Explanation:

The statement the teacher makes is “The set of whole numbers is closed under multiplication, but the set of irrational numbers is not closed under multiplication.”

For a set of numbers to closed under multiplication means that, if we multiply any two numbers from the set, the result is also a number that belongs to the same set.

For example, the set of whole numbers are : {0,1,2,3,4,5,6,...}

`0* 1=0`

`2* 3=6`

`5* 6=30`

Observe that, the results we are getting are also whole numbers.

Also the irrational numbers are not closed under multiplication.

This means that we can multiply some two irrational numbers and get a rational number.

For example:
`√(2)* √(8)=√(8*2)=√(16)=4`

`√(2), and\:√(8)` are irrational but their product is 4 which is rational