Rodney Nillsen

The golden ratio is the number In the regular pentagon in Figure 1, the ratio of the length of a diagonal of the pentagon to the length of a side equals the golden ratio. In the argument below, we show that if the golden ratio is rational, we are led to a contradiction - that is, a logical impossibility. Consequently, the golden ratio is not a rational number. It follows that the square root of 5 is not a rational number either.

DEFINITIONS. If x and y are two numbers, x is called a multiple of y if there is an integer p such that x=py. A number x is called rational if there are integers p and q with q non-zero such that x=p/q. A number that is not rational is called irrational.
Note that if x and z are both multiples of y, then x-z is also a multiple of y.

Important fact. If x and y are non-zero numbers with x/y rational, then x and y are both multiples of some number r. That is, there is a number r and integers p and q such that x=pr and y=qr.
Proof of this fact. Observe that as x/y is rational, there are integers p, q such that x/y=p/q, and p and q are non-zero. Then x/p=y/q, and put r=x/p=y/q. Then, r is non-zero, x=pr and y=qr. That is, both x and y are multiples of r.

Consider the regular pentagon in Figure 2, together with the diagonal BE. Note that BE is parallel to CD. In Figure 2, assume that the fraction BE/ CD is rational. Then, using the fact above, there is a number r such that : BE is a multiple of r , and CD is a multiple of r.

In Figure 3, BCDY is a parallelogram. So, CD=BY. As CD is a multiple of r, BY is a multiple of r. So, BY and BE are multiples of r, so EY is a multiple of r, because EY=BE-BY. But EY and BX are equal, and so EY and BX are multiples of r. Hence, BE, BX and EY are multiples of r. Because XY=BE-BX-EY, we nowhave: XY is a multiple of r, and BX is a multiple of r.

In Figure 4, BPRX is a parallelogram. So, BX=PR. As BX is a multiple of r, from above, PRis a multiple of r. We now have: XY is a multiple of r, and PR is a multiple of r.

To arrive at Figure 5, we used the regular pentagon ABCDE to construct the smaller regular pentagon XPQRY as n Figure 4. We have seen above that if BE and CD are multiples of r, then PR and XY are also multiples of r. This argument and construction can now be repeated, so that : when we obtain the new pentagon from the old one, both the sides and the diagonals of this new pentagon are also multiples of r.

Figure 6 illustrates the first few regular pentagons in an infinite sequence of pentagons, each one constructed from the preceding one as described above. The above argument shows that all the sides and diagonals of all pentagons in the sequence are multiples of r. Thus, each side of each pentagon in the sequence of pentagons has a length that is at least r. But, this is impossible, because the pentagons are decreasing by a fixed ratio in size at each stage and eventually one of them will have a side of length less than r . Thus, if we assume that the ratio of the length of the diagonal of a regular pentagon to the length of the side of a regular pentagon is a rational number, we are led to a situation that is impossible. Our conclusion is: The assumption that the ratio is rational is incorrect. So, the ratio of the length of a diagonal of a regular pentagon to the side of the pentagon is irrational.

Comments

The discovery of irrational numbers or "incommensurable quantities" is sometimes attributed to the Pythagorean philosopher Hippasos of Metapont (about 450 BCE). and it seems possible that he did this by some version of the argument presented above (see for example Michael Lahanas' page The irrationality of the Pentagon and the Pentagram). The above argument is visual rather than formal, and the review article Visual explanations by Bill Casselman has some discussion on the nature of visual proof. His article has also been influential on the presentation above.

Rodney Nillsen, February 2007

go to home page