**Visual Explanations in Mathematics**

## 4. Proofs of ancillary facts

1. *In a regular pentagon, the ratio of diagonal to side is the
Golden Mean 1.618033...*
Proof: Use the notation from the
incommensurability proof. The ratio we are interested in is
`d/s`. This ratio must be the same in the smaller regular
pentagon, where it becomes `d*/s*`. Substituting into
the equation `d/s = d*/s*` the
expressions for `d*` and `s*` in terms of
`d` and `s` yields:

d d - s
--- = -------- .
s 2s - d

Multiplying out and regrouping gives:
d^{2} - sd - s^{2} = 0.

Dividing through by `s`^{2} leaves us with:
d^{2} d
--- - --- - 1 = 0.
s^{2} s

So `d/s` is the positive root of the equation
`x`^{2} - x - 1 = 0, i.e. the Golden
Mean.

2. *A number is rational if and only if it is commensurable with 1.*

Proof: If a number `x` is commensurable with `1`, that means
there exists a number `h` which is contained exactly
a whole number of times in `1` and in `x`. Suppose
it is contained `q` times in `1` and `p`
times in `x`. Then `h = 1/q` and `x = p/q`,
so `x` is rational. Conversely if `x` is
rational, say `x = p/q` with `p` and `q`
integers, then taking `h = 1/q` shows that `x`
and `1` are commensurable.

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*

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