A Neat and Simple Formula for the Golden Ratio

This week, I thought I would talk a little bit about the golden ratio. The golden ratio is a topic in Fibonacci numbers, but this post won’t be involving them, so I did not bother saving it for a Fibonacci day.

There is a formula for the golden ratio, as follows:

(1 + √5)/2

However, there is another formula that has a little more beauty in it.

1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...

I’d say that is a little cooler than the previous formula. However, this is a pretty difficult statement to prove what this is equal to. However, we can figure it out.

First, let’s set this formula equal to x.

x = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...

Ignore the 1 + 1 at the beginning. What do we have?

1 + 1/(1 + 1/(1 + 1/(1 + 1/...

If you’ll notice, this is the same exact thing as before, since they are both an infinite series. Therefore, the equation can be rewritten as:

x = 1 + 1/x

Now, we can multiply through by x.

x(x) = x(1 + 1/x)

x^2 = x + 1

Now, we will subtract x + 1 from both sides to get:

x^2 - x - 1 = 0

We have just turned that infinite series into a solvable equation! A little while ago, we talked about a formula called the quadratic formula, which is used to solve equations just like this one. The formula is:

(-b ± √(b^2 - 4ac))/2a

With a, b, and c being the first, second, and third coefficient. So, let’s plug them in and see what we get.

(-(-1) ± √((-1)^2 - 4(1)(-1)))/2(1)

(1 ± √(1 + 4))/2

(1 ± √5)/2

We are in home stretch. Now, here is where logic will come in. One answer here is correct, and one is called an extraneous solution (based on logic, it is not a valid solution). Since we are adding a positive quantity to one, we know the solution must be positive. Therefore, (1 - √5)/2 would not be a valid answer. So, we are left with:

x = (1 + √5)/2

So, this proves the golden ratio equal to that top number.

The golden ratio’s formula is very cool, but there are more continued fractions such as this one. For instance, pi (or as I now call it: half-tau) has a formula as well. Check it out:

π = 3 + 1/(6 + 9/(6 + 25/(6 + 49/(6 + 81/...

This one isn’t quite as cool, but still very interesting.

Bonus: A month or so ago, I posted a puzzle involving light bulbs. If you figured it out, great. If you haven’t, here is the answer.

Turn on two of the light switches and wait for a little while. After some time, turn off one of them, and leave the other one on. Then, go upstairs.

You will see the on light bulb, which you can immediately match up. For the two off bulbs, feel them and see which has more heat. The one with more heat is the one you turned on and then turned off (it would still have heat from it being lit up), and the one with less heat was never turned on.