Gilded Ratios


I may have mentioned that I regard the Golden Ratio as a lot of bunk. If I haven’t, allow me to mention: the Golden Ratio is a lot of bunk. I concede it’s a cute number. I found it compelling when I first had a calculator that let me use the last answer for a new operation. You can pretty quickly find that 1.618033 (etc, and the next digit is a 9 by the way) has a reciprocal that’s 0.618033 (etc).

There’s no denying that. And there’s no denying that’s a neat pattern. But it is not some aesthetic ideal. When people evaluate rectangles that “look best” they go to stuff that’s a fair but not too much wider in one direction than the other. But people aren’t drawn to 1.618 (etc) any more reliably than they like 1.6, or 1.8, or 1.5, or other possible ratios. And it is not any kind of law of nature that the Golden Ratio will turn up. It’s often found within the error bars of a measurement, but so are a lot of numbers.

The Golden Ratio is an irrational number, but basically all real numbers are irrational except for a few peculiar ones. Those peculiar ones happen to be the whole numbers and the rational numbers, which we find interesting, but which are the rare exception. It’s not a “transcendental number”, which is a kind of real number I don’t want to describe here. That’s a bit unusual, since basically all real numbers are transcendental numbers except for a few peculiar ones. Those peculiar ones include whole and rational numbers, and square roots and such, which we use so much we think they’re common. But not being transcendental isn’t that outstanding a feature. The Golden Ratio is one of those strange celebrities who’s famous for being a celebrity, and not for any actual accomplishment worth celebrating.

I started wondering: are there other Golden-Ratio-like numbers, though? The title of this essay gives what I suppose is the best name for this set. The Golden Ratio is interesting because its reciprocal — 1 divided by it — is equal to it minus 1. Is there another number whose reciprocal is equal to it minus 2? Another number yet whose reciprocal is equal to it minus 3?

So I looked. Is there a number between 2 and 3 whose reciprocal is it minus 2? Certainly there is. How do I know this?

Let me call this number, if it exists, x. The reciprocal of x is the number 1/x. The number x minus 2 is the number x – 2. We’ll pick up the pace in a little bit. Now imagine trying out every single number from 2 to 3, in order. The reciprocals 1/x start out at 1/2 and drop to 1/3. The subtracted numbers start out at 0 and grow to 1. There’s no gaps or sudden jumps or anything in either the reciprocals or the subtracted numbers. So there must be some x for which 1/x and x – 2 are the same number.

In the trade we call that an existence proof. It shows there’s got to be some answer. It doesn’t tell us much about what the answer is. Often it’s worth looking for an existence proof first. In this case, it’s probably overkill. But you can go from this to reasoning that there have to be Golden-Like-Ratio numbers between any two counting numbers. So, yes, there’s some number between 2,038 and 2,039 whose reciprocal is that number minus 2,038. That’s nice to know.

So what is the number that’s two more than its reciprocal? That’s whatever number or numbers make true the equation \frac{1}{x} = x - 2 . That’s straightforward to solve. Multiply both sides by x, which won’t change whether the equation is true unless x is zero. (And x can’t be zero, or else we wouldn’t talk of 1/x except in hushed, embarrassed whispers.) This gets an equivalent equation 1 = x^2 - 2x . Subtract 1 from both sides, and we get 0 = x^2 - 2x - 1 and we’re set up to use the quadratic formula. The answer will be x = \left(\frac{1}{2}\right)\cdot\left(2 + \sqrt{2^2 + 4}\right) . The answer is about 2.414213562373095 (and on). (No, \left(\frac{1}{2}\right)\cdot\left(2 - \sqrt{2^2 + 4}\right) is not an answer; it’s not between 2 and 3.)

The number that’s three more than its reciprocal? We’ll call that x again, trusting that we remember this is a different number with the same name. For that we need to solve \frac{1}{x} = x - 3 and that turns into the equation 0 = x^2 - 3x - 1 . And so x = \left(\frac{1}{2}\right)\cdot\left(3 + \sqrt{3^2 + 4}\right) and so it’s about 3.30277563773200. Yes, there’s another possible answer we rule out because it isn’t between 3 and 4.

We can do the same thing to find another number, named x, that’s four more than its reciprocal. That starts with \frac{1}{x} = x - 4 and gets eventually to x = \left(\frac{1}{2}\right)\cdot\left(4 + \sqrt{4^2 + 4}\right) or about 4.23606797749979. We could go on like this. The number x that’s 2,038 more than its reciprocal is x = \left(\frac{1}{2}\right)\cdot\left(2038 + \sqrt{2038^2 + 4}\right) about 2038.00049082160.

If your eyes haven’t just slid gently past the equations you noticed the pattern. Suppose instead of saying 2 or 3 or 4 or 2038 we say the number b. b is some whole number, any that we like. The number whose reciprocal is exactly b less than it is the number x that makes true the equation \frac{1}{x} = x - b . And that leads to the finding the number that makes the equation x = \left(\frac{1}{2}\right)\cdot\left(b + \sqrt{b^2 + 4}\right) true.

And, what the heck. Here’s the first twenty or so gilded numbers. You can read this either as a list of the numbers I’ve been calling x — 1.618034, 2.414214, 3.302776 — or as an ordered list of the reciprocals of x — 0.618034, 0.414214, 0.302276 — as you like. I’ll call that the gilt; you add it to the whole number to its left to get that a number that, cutely, has a reciprocal that’s the same after the decimal.

I did think about including a graph of these numbers, but the appeal of them is that you can take the reciprocal and see digits not changing. A graph doesn’t give you that.

Some Numbers With Cute Reciprocals
Number Gilt
1 .618033989
2 .414213562
3 .302775638
4 .236067977
5 .192582404
6 .162277660
7 .140054945
8 .123105626
9 .109772229
10 .099019514
11 .090169944
12 .082762530
13 .076473219
14 .071067812
15 .066372975
16 .062257748
17 .058621384
18 .055385138
19 .052486587
20 .049875621

None of these are important numbers. But they are pretty, and that can be enough on a quiet day.

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