φ and Fibonacci
by Emily Kennedy


Mathematicians have long been fascinated by a particular number, called the Golden Ratio. For an excellent read about the properties and surprises of the Golden Ratio, pick up a copy of this book by Mario Livio.

There are a lot of equivalent ways to define the golden ratio; let's use the following definition:

For any segment AB, there is a unique point C on AB such that
.
For such a point C, the ratio (or the equivalent ratio )
is , the Golden Ratio.

Let's find the value of !

Let x be the length of , and let y be the length of .
Then the length of is x + y, so

By the quadratic formula, we get:

Note that both AC and BC have positive lengths (of course), so

But , so we must have

We have found two properties of :
  and  

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