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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?

Michael Lee
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    Well, since $\phi$ is irrational, it means that $a$ and $b$ can't both be integers. A fraction is a ratio of $2$ integers. – Peter Woolfitt Jan 11 '15 at 23:13
  • $\phi=[1;1,1,1,1,1,1,1,1,1,1,1,\ldots]$ – jimbo Jan 11 '15 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. – Mufasa Jan 11 '15 at 23:15
  • @jimbo While correct, what does this help the questioner? ^^ – AlexR Jan 11 '15 at 23:17
  • @jimbo While correct, I doubt that that notation is familiar to the original questioner :) – Alan Jan 11 '15 at 23:18
  • For anyone who likes continued fractions, in some sense $\phi$ is the most irrational number, because its continued fraction expansion uses the smallest values possible. (A rational number has a finite number of integers in its continued fraction expansion, effectively followed by an $\infty$---so not small.) This is why it shows up in seed heads. If seeds are released at rational angles relative to a whole revolution they will end up clustered too close. It's this sense of being far away from rational that causes $\phi$ to be a good proportion of a revolution for seeds to be released. – 2'5 9'2 Jan 11 '15 at 23:19
  • @jimbo While correct, previous to this I wasn't familiar with the notation you're using. – hjhjhj57 Jan 11 '15 at 23:19
  • I just can't believe I went all these years without knowing an irrational number cannot be expressed as a fraction of integers. (No jokes please) The belief that all numbers can be represented by such a fraction must have been important to the ancient mathematicians. The discovery of irrational numbers (Hippasus?) must have really freaked them out. – Michael Lee Jan 12 '15 at 16:30

2 Answers2

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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 $a=\frac a 1$

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

Hagen von Eitzen
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