Question

Here are two expressions with the missing number

Answer 1

`√(3)` for each blank

Explanation:

The value 3.2 is rational because 3.2 = 32/10 = 16/5. We can rewrite 3.2 as a fraction of two integers, which is what makes it rational.

The same can be said about 0.232323... where the "23" goes on forever. It turns out that 0.232323... = 23/99; therefore, 0.232323... is rational.

The fraction 2/3 is also rational because it is a ratio of two integers.

The only item of the list that isn't rational, aka irrational, is
`√(3)`. We cannot write this as a ratio of integers. We can get approximations, but we can't get the exact value itself.

A rule to write down is: irrational*rational = irrational. This means something like
`2√(3)` is irrational because 2 is rational,
`√(3)` is irrational. The proof is a bit lengthy to write here. But let me know if you need to see the proof.

Also, when adding an irrational value to a rational one, we get an irrational answer. The template is: irrational + rational = irrational

This is why we put
`√(3)` into each blank to make both results irrational.