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

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?

• Well, since $\phi$ is irrational, it means that $a$ and $b$ can't both be integers. A fraction is a ratio of $2$ integers. Jan 11, 2015 at 23:13
• $\phi=[1;1,1,1,1,1,1,1,1,1,1,1,\ldots]$ Jan 11, 2015 at 23:14
• Indeed, you can write $\varphi=\frac{1+\sqrt{5}}{2}$ but this does not make it rational as the fraction has an irrational numerator in this case. Jan 11, 2015 at 23:15
• @jimbo While correct, what does this help the questioner? ^^ Jan 11, 2015 at 23:17
• @jimbo While correct, I doubt that that notation is familiar to the original questioner :)
– Alan
Jan 11, 2015 at 23:18

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 $a=\frac a 1$

• That is helpful because I thought a and or b can be any real number. I greatly appreciate the help from all of you and it makes much more sense now! Jan 11, 2015 at 23:37
• My pleasure! Good learning.
– Alan
Jan 11, 2015 at 23:38
• Note that if they could be any real number then $a = \frac{a}{1}$ implies everything is rational.
– jxnh
Jan 12, 2015 at 5:32

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