Question

What number produces an irrational number when added to 1/4?

Answer 1

Any irrational number, e.g. `sqrt(2)`

Explanation:

`x + 1/4` is irrational if and only if `x` is irrational.

Equivalently, `x + 1/4` is rational if and only if `x` is rational.

To prove this we can proceed as follows:

First suppose that `x+1/4` is rational.

Then there are some integers `p, q`, with `q > 0` such that:

`x+1/4 = p/q`

Subtracting `1/4` from both sides, this becomes:

`x = p/q - 1/4 = (4p-q)/(4q)`

which is rational.

Conversely, if `x` is rational, then there are integers `m, n` with `n > 0` such that `x = m/n` and we find:

`x+1/4 = m/n+1/4 = (4m+n)/(4n)`

which is also rational.