What do we mean when we say an irrational number can't be expressed as a fraction?

Question

An irrational number is one such that it cannot be expressed by a fraction, but consider the definition of the Golden Ratio.

Two line segments, call one a and the other b, are said to be of the Golden Ratio if: $${{a + b} \over a} = {a \over b} = \varphi $$

How can,

$${a \over b} = \varphi $$

be the case if an irrational number cannot be expressed as a fraction?

Answer 1

Your definition of irrational is incomplete. A number is irrational if it cannot be expressed in terms of $\frac a b$ where both $a$ and $b$ are INTEGERS ($b\ne 0$). In this case, the $a$ and $b$ are not simultaneously integers, so it is irrational.

Edit for further clarity:

If the restriction of "integers" was removed, then every number would be "rational", because $a=\frac a 1$

Answer 2

An irrational number is one that cannot be expressed by a fraction of integers.