 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.

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .

Search Feature Column

Feature Column at a glance