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.