visual4 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:
d2 - sd - s2 = 0.
Dividing through by s2 leaves us with:
d2 d --- - --- - 1 = 0. s2 s
So d/s is the positive root of the equation x2 - 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.