2.2: The Golden Ratio Φ

The number \(\Phi\) is known as the golden ratio. Two positive numbers \(x\) and \(y\), with \(x>y\), are said to be in the golden ratio if the ratio between the sum of those numbers and the larger one is the same as the ratio between the larger one and the smaller; that is,

\[\frac{x+y}{x}=\frac{x}{y} \nonumber \]

Solution of (2.2.1) yields \(x / y=\Phi\). In some well-defined way, \(\Phi\) can also be called the most irrational of the irrational numbers.

To understand why \(\Phi\) has this distinction as the most irrational number, we need first to understand continued fractions. Recall that a rational number is any number that can be expressed as the quotient of two integers, and an irrational number is any number that is not rational. Rational numbers have finite continued fractions; irrational numbers have infinite continued fractions.

A finite continued fraction represents a rational number \(x\) as

\[x=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{\ddots+\frac{1}{a_{n}}}}} \nonumber \]

where \(a_{1}, a_{2}, \ldots, a_{n}\) are positive integers and \(a_{0}\) is any integer. The convenient shorthand form of \((2.2.2)\) is

\[x=\left[a_{0} ; a_{1}, a_{2}, \ldots, a_{n}\right] \nonumber \]

If \(x\) is irrational, then \(n \rightarrow \infty\).

Now for some examples. To construct the continued fraction of the rational number \(x=3 / 5\), we can write

\[\begin{aligned} 3 / 5 &=\frac{1}{5 / 3}=\frac{1}{1+2 / 3} \\[4pt] &=\frac{1}{1+\frac{1}{3 / 2}}=\frac{1}{1+\frac{1}{1+1 / 2}} \end{aligned} \nonumber \]

which is of the form \((2.2.2)\), so that \(3 / 5=[0 ; 1,1,2]\).

To construct the continued fraction of the irrational number \(\sqrt{2}\), we can make use of a trick and write

\[\begin{aligned} \sqrt{2} &=1+(\sqrt{2}-1) \\[4pt] &=1+\frac{1}{1+\sqrt{2}} \end{aligned} \nonumber \]

We now have a recursive definition that can be continued as

\[\begin{aligned} \sqrt{2} &=1+\frac{1}{1+\left(1+\frac{1}{1+\sqrt{2}}\right)} \\[4pt] &=1+\frac{1}{2+\frac{1}{1+\sqrt{2}}} \end{aligned} \nonumber \]

and so on, which yields the infinite continued fraction

\[\sqrt{2}=[1 ; \overline{2}] \nonumber \]

Another example we will use later is the continued fraction for \(\pi\), whose first few terms can be calculated from

\[\begin{aligned} \pi &=3+0.14159 \ldots \\[4pt] &=3+\frac{1}{7.06251 \ldots} \\[4pt] &=3+\frac{1}{7+\frac{1}{15.99659 \ldots}} \end{aligned} \nonumber \]

and so on, yielding the beginning sequence \(\pi=[3 ; 7,15, \ldots]\). The historically important first-order approximation is given by \(\pi=[3 ; 7]=22 / 7=3.142857 \ldots\), which was already known by Archimedes in ancient times.

Finally, to determine the continued fraction for the golden ratio \(\Phi\), we can write

\[\Phi=1+\frac{1}{\Phi} \nonumber \]

which is another recursive defintion that can be continued as

\[\Phi=1+\frac{1}{1+\frac{1}{\Phi}} \nonumber \]

and so on, yielding the remarkably simple form

\[\Phi=[1 ; \overline{1}] . \nonumber \]

Because the trailing \(a_{i}\) ’s are all equal to one, the continued fraction for the golden ratio (and other related numbers with trailing ones) converges especially slowly. Furthermore, the successive rational approximations to the golden ratio are just the ratio of consecutive Fibonacci numbers, that is, \(1 / 1,2 / 1,3 / 2,5 / 3\), etc.. Because of the very slow convergence of this sequence, we say that the golden ratio is most difficult to approximate by a rational number. More poetically, the golden ratio has been called the most irrational of the irrational numbers.

Because the golden ratio is the most irrational number, it has a way of appearing unexpectedly in nature. One well-known example is the florets in a sunflower head, which we discuss in the next section.