Question

Here are two expressions with a mixing number:

Answer 1

An irrational number is a real number that can't be written as a fraction: since all rational numbers (
`\mathbb{Q}`) are real numbers (
`\mathbb{R}`), the irrational numbers are all elements in
`\mathbb{R}` that don't belong to
`\mathbb{Q}`.

The rational numbers are closed with respect to sum and multiplication, which means that the sum/product of two rational numbers is a rational number.

Note that, of all the options, only
`√(3)` is irrational, because the other options are

`3 = (3)/(1),\quad 2 =(2)/(1),\quad 0.\overline{23} = (23)/(99)`

so they all can be written as fraction. This means that

`-(5)/(9) * 3,\quad -(5)/(9) * 2,\quad -(5)/(9) *(23)/(99)`

are all rational numbers, because they are the multiplication of two rational numbers.

So, your only hope to get an irrational number is to multiply

`-(5)/(9)* √(3)`

Expression 2 works exactly in the same way, because everything we said about the product is true for the sum as well.