Silver-Leafed Numbers

In a comment on my “Gilded Ratios” essay fluffy wondered about a variation on the Golden and Golden-like ratios. What’s interesting about the Golden Ratio and similar numbers is that their reciprocal — one divided by them — is a whole number less than the original number. That is, 1 divided by 1.618(etc) is 0.618(etc), which is 1 less than the original number. 1 divided by 2.414(etc) is 0.414(etc), exactly 2 less than the original 2.414(etc). 1 divided by 3.302(etc) is 0.302(etc), exactly 3 less than the original 3.302(etc).

fluffy wondered about a variation. Is there some number x that’s exactly 2 less than 2 divided by x? Or a (presumably) differently number that’s exactly 3 less than 3 divided by it? Yes, there is.

Let me call the whole number difference — the 1 or 2 or 3 or so on, referred to above — by the name b. And let me call the other number — the one that’s b less than b divided by it — by the name x. Then a number x, for which b divided by x is exactly b less than itself, makes true the equation \frac{b}{x} = x - b . This is slightly different from the equation used last time, but not very different. Multiply both sides by x, which we know not to be zero, and we get a polynomial.

Yes, quadratic formula, I see you waving your hand in the back there. And you’re right. There are two x’s that will make that equation true. The positive one is x = \frac12\left( b + \sqrt{b^2 + 4b} \right) . The negative one you get by changing the + sign, just before the square root, to a – sign, but who cares about that root? Here’s the first several of the (positive) silver-leaf ratios:

Some More Numbers With Cute Reciprocals
Number Silver-Leaf
1 1.618033989
2 2.732050808
3 3.791287847
4 4.828427125
5 5.854101966
6 6.872983346
7 7.887482194
8 8.898979486
9 9.908326913
10 10.916079783
11 11.922616289
12 12.928203230
13 13.933034374
14 14.937253933
15 15.940971508
16 16.944271910
17 17.947221814
18 18.949874371
19 19.952272480
20 20.954451150

Looking over those hypnotic rows of digits past the decimal inspires thoughts. The part beyond the decimal keeps rising, closer and closer to 1. Does it ever get past 1? That is, might (say) the silver-leaf number that’s 2,038 more than its reciprocal be 2,039.11111 (or something)?

No, it never does. There are a couple of ways to prove that, if you feel like. We can take the approach that’s easiest (to my eyes) to imagine. It takes a little algebraic grinding to complete. That is to look for the smallest number b for which the silver-leaf number, \frac12\left(b + \sqrt{b^2 + 4b}\right) , is larger than b + 1 . Follow that out and you realize that it’s any value of b for which 0 is greater than 4. Logically, therefore, we need to take b into a private room and have a serious talk about its job performance, what with it not existing.

A harder proof to imagine working out, but that takes no symbol manipulation, comes from thinking about these reciprocals. Let’s imagine we had some b for which its corresponding silver-leaf number x is more than b + 1. Then, x – b has to be greater than 1. But if x is greater than 1, then its reciprocal has to be less than 1. We again have to talk with b about how its nonexistence is keeping it from doing its job.

Are there other proofs? Most likely. I was satisfied by this point, and resolved not to work on it more until the shower. Updates after breakfast, I suppose.


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.

A Summer 2015 Mathematics A To Z: fallacy


Mathematics is built out of arguments. These are normally logical arguments, sequences of things which we say are true. We know they’re true because either they start from something we assume to be true or because they follow from logical deduction from things we assumed were true. Even calculations are a string of arguments. We start out with an expression we’re interested in, and do things which change the way it looks but which we can prove don’t change whether it’s true.

A fallacy is an argument that isn’t deductively sound. By deductively sound we mean that the premises we start with are true, and the reasoning we follow obeys the rules of deductive logic (omitted for clarity). if we’ve done that, then the conclusion at the end of the reasoning is — and must be — true.

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What We Can Say About Nonexistent Things

The modern interpretation of what we mean by a statement like “all unicorns are one-horned animals” is that we aren’t making the assertion that any unicorns exist. If any did happen to exist, sure, they’d be one-horned animals, if our proposition is true, but we’re reserving judgement about whether they do exist. If we don’t like the way the natural-language interpretation of the proposition leads us, we might be satisfied by saying it’s equivalent to saying, “there are no non-one-horned animals which are unicorns”, and that doesn’t feel quite like it claims unicorns exist. You might not even come away feeling there ought to be non-one-horned animals from that sentence alone.

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What Lewis Carroll Says Exists That I Don’t

I borrowed from the library Symbolic Logic, a collection of an elementary textbook — intended for children, and more fun than usual because of that — on logic by Lewis Carroll, combined with notes and manuscript pages which William Warren Bartley III found toward the second volume in the series. The first part is particularly nice since it’s text that not only was finished in Carroll’s life but went through several editions so he could improve the unclear parts. In case I do get to teaching a new logic course I’ll have to plunder it for examples as well as for this rather nice visual representation Carroll used for sorting out what was implied by a set of propositions regard “All (something) are (something else)” and “Some (something) are (this)” and “No (something) are (whatnot)”. It’s not quite Venn diagrams, although you can see them from there. Oddly, Carroll apparently couldn’t; there’s a rather amusing bit in the second volume where Carroll makes Venn diagrams out to be silly because you can make them terribly complicated.

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