Golden Ratio: Proof

We are looking for a way to calculate the value of , the Golden Ratio. All we know about is that it is the ratio of the length and height of a Golden Rectangle.

What do we know about a Golden Rectangle? Well, we know that it is a rectangle which, when squared, leaves behind another rectangle of the same proportions. That is, the ratios of the lengths and widths of the rectangles is the same.

Based on this knowledge, we can set up a proportion, like so:

Before going any further, let's eliminate one of the variables by assuming that the short side of the larger rectangle (the value x) is equal to 1. As we are only looking for the ratio of the sides, this assumption will not alter the problem in any way. Our proportion now becomes:

By cross-multiplying, we obtain:

Some simple algebraic moving about produces the quadratic equation

The next step is to solve for the value y by applying the quadratic formula. When we do, we obtain the following values:

We can discard the second value because it is negative, and lengths of polygons can not be negative. Now we are left with:

All this so far has been done to find the value of , which is defined as y/x. We have found the value of y, and we said before we started that x was equal to 1. Therefore, the value of y/x is the same as the value of y (because y/1 = y), so:

Remember at least the first few digits of this decimal approximation of , as it will become important in future lessons. Note that this is only an approximation, because is actually an irrational number, meaning that it can not be expressed as the ratio of two whole numbers.

You can try this method again using different values of x, and you will always come out with the same value for .