Lesson

Golden ratio

The golden ratio denoted here by is also called the golden mean or section. It has the value, = 1.61803398.... We show how this comes about in two ways. One is by an algebraic relation. Suppose that a line segment is cut into two pieces of length a and b, and we have shown here a is longer than b. Clearly, the length of the original line segment is a+b.

We now form two ratios and . The first is ratio of the whole length to longer piece a, and the second ratio is longer piece a to shorter piece b. We now ask, when are the two ratios equal? This is an algebra question that can be solved by equating the two ratios

=

(1)

First of all, by dividing each and every term of equation (1) by a, we get = 1 + . So, denoting the ratio by = , we end up with

On the other hand, dividing each and every term of equation (1) by b, we get , which by writing = becomes a little more complicated expression

But you know that equation (2) and (3) are same, just written differently.

Now, by multiplying both sides of equation (3) by we get a quadratic equation

Using the quadratic root formula, we find two roots of equation (4); the first root is 1.618 and the second root is-0.618. The first root is the ratio = that we are looking for, and we ignore the second root because it is a negative number. We can easily check to see if = 1.618 is the right answer. With a = 1.618b, the two ratios and 1.618 are indeed the same. (Project a – The decimal digits of by continued fractions) Now, by subtracting 1 from both sides of equation (2) we get

This means that - 1= 0.61803… is same as its own reciprocal = 0.61803… In other words, and have the same decimal digits. For instance, try for = 3 in (5). Then the left-hand side 3 -1 = 2 and the right-hand side 1/3 = 0.333... are not certainly equal.

The second way to define the golden ratio is by a geometric relation. Consider the square A-B-C-D in figure 1, which has the side of length 1 in whatever the unit you choose. For instance, it may be 1 meter or 1 yard. We denote by E the midpoint of side A-D. Draw a circular arc with radius E-C and denote by F where the arc intersects the extended line segment of A-D, as shown in figure 1. Then the line segment A-F will have length 1.618… (Project b – Show the length A-F is ?)

Figure 1. Geometric construction of golden ratio

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