What Is The Logarithm of a Negative Number?

Learning of imaginary numbers, things created to be the square roots of negative numbers, inspired me. It probably inspires anyone who’s the sort of person who’d become a mathematician. The trick was great. I wondered could I do it? Could I find some other useful expansion of the number system?

The square root of a complex-valued number sounded like the obvious way to go, until a little later that week when I learned that’s just some other complex-valued numbers. The next thing I hit on: how about the logarithm of a negative number? Couldn’t that be a useful expansion of numbers?

No. It turns out you can make a sensible logarithm of negative, and complex-valued, numbers using complex-valued numbers. Same with trigonometric and inverse trig functions, tangents and arccosines and all that. There isn’t anything we can do with the normal mathematical operations that needs something bigger than the complex-valued numbers to play with. It’s possible to expand on the complex-valued numbers. We can make quaternions and some more elaborate constructs there. They don’t solve any particular shortcoming in complex-valued numbers, but they’ve got their uses. I never got anywhere near reinventing them. I don’t regret the time spent on that. There’s something useful in trying to invent something even if it fails.

One problem with mathematics — with all intellectual fields, really — is that it’s easy, when teaching, to give the impression that this stuff is the Word of God, built into the nature of the universe and inarguable. It’s so not. The stuff we find interesting and how we describe those things are the results of human thought, attempts to say what is interesting about a thing and what is useful. And what best approximates our ideas of what we would like to know. So I was happy to see this come across my Twitter feed:

The links to a 12-page paper by Deepak Bal, Leibniz, Bernoulli, and the Logarithms of Negative Numbers. It’s a review of how the idea of a logarithm of a negative number got developed over the course of the 18th century. And what great minds, like Gottfried Leibniz and John (I) Bernoulli argued about as they find problems with the implications of what they were doing. (There were a lot of Bernoullis doing great mathematics, and even multiple John Bernoullis. The (I) is among the ways we keep them sorted out.) It’s worth a read, I think, even if you’re not all that versed in how to calculate logarithms. (but if you’d like to be better-versed, here’s the tail end of some thoughts about that.) The process of how a good idea like this comes to be is worth knowing.

Also: it turns out there’s not “the” logarithm of a complex-valued number. There’s infinitely many logarithms. But they’re a family, all strikingly similar, so we can pick one that’s convenient and just use that. Ask if you’re really interested.

Reading the Comics, June 25, 2016: What The Heck, Why Not Edition

I had figured to do Reading the Comics posts weekly, and then last week went and gave me too big a flood of things to do. I have no idea what the rest of this week is going to look like. But given that I had four strips dated before last Sunday I’m going to err on the side of posting too much about comic strips.

Scott Metzger’s The Bent Pinky for the 24th uses mathematics as something that dogs can be adorable about not understanding. Thus all the heads tilted, as if it were me in a photograph. The graph here is from economics, which has long had a challenging relationship with mathematics. This particular graph is qualitative; it doesn’t exactly match anything in the real world. But it helps one visualize how we might expect changes in the price of something to affect its sales. A graph doesn’t need to be precise to be instructional.

Dave Whamond’s Reality Check for the 24th is this essay’s anthropomorphic-numerals joke. And it’s a reminder that something can be quite true without being reassuring. It plays on the difference between “real” numbers and things that really exist. It’s hard to think of a way that a number such as two could “really” exist that doesn’t also allow the square root of -1 to “really” exist.

And to be a bit curmudgeonly, it’s a bit sloppy to speak of “the square root of negative one”, even though everyone does. It’s all right to expand the idea of square roots to cover stuff it didn’t before. But there’s at least two numbers that would, squared, equal -1. We usually call them i and -i. Square roots naturally have this problem,. Both +2 and -2 squared give us 4. We pick out “the” square root by selecting the positive one of the two. But neither i nor -i is “positive”. (Don’t let the – sign fool you. It doesn’t count.) You can’t say either i or -i is greater than zero. It’s not possible to define a “greater than” or “less than” for complex-valued numbers. And that’s even before we get into quaternions, in which we summon two more “square roots” of -1 into existence. Octonions can be even stranger. I don’t blame 1 for being worried.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is a pleasant bit of pop-mathematics debunking. I’ve explained in the past how I’m a doubter of the golden ratio. The Fibonacci Sequence has a bit more legitimate interest to it. That’s sequences of numbers in which the next term is the sum of the previous two terms. The famous one of that is 1, 1, 2, 3, 5, 8, 13, 21, et cetera. It may not surprise you to know that the Fibonacci Sequence has a link to the golden ratio. As it goes on, the ratio between one term and the next one gets close to the golden ratio.

The Harmonic Series is much more deeply weird. A series is the number we get from adding together everything in a sequence. The Harmonic Series grows out of the first sequence you’d imagine ever adding up. It’s 1 plus 1/2 plus 1/3 plus 1/4 plus 1/5 plus 1/6 plus … et cetera. The first time you hear of this you get the surprise: this sum doesn’t ever stop piling up. We say it ‘diverges’. It won’t on your computer; the floating-point arithmetic it does won’t let you add enormous numbers like ‘1’ to tiny numbers like ‘1/531,325,263,953,066,893,142,231,356,120’ and get the right answer. But if you actually added this all up, it would.

The proof gets a little messy. But it amounts to this: 1/2 plus 1/3 plus 1/4? That’s more than 1. 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12? That’s also more than 1. 1/13 + 1/14 + 1/15 + et cetera up through + 1/32 + 1/33 + 1/34 is also more than 1. You need to pile up more and more terms each time, but a finite string of these numbers will add up to more than 1. So the whole series has to be more than 1 + 1 + 1 + 1 + 1 … and so more than any finite number.

That’s all amazing enough. And then the series goes on to defy all kinds of intuition. Obviously dropping a couple of terms from the series won’t change whether it converges or diverges. Multiplying alternating terms by -1, so you have (say) 1 – 1/2 + 1/3 – 1/4 + 1/5 et cetera produces something that looks like it converges. It equals the natural logarithm of 2. But if you take those terms and rearrange them, you can produce any real number, positive or negative, that you want.

And, as Weinersmith describes here, if you just skip the correct set of terms, you can make the sum converge. The ones with 9 in the denominator will be, then, 1/9, 1/19, 1/29, 1/90, 1/91, 1/92, 1/290, 1/999, those sorts of things. Amazing? Yes. Absurd? I suppose so. This is why mathematicians learn to be very careful when they do anything, even addition, infinitely many times.

John Deering’s Strange Brew for the 25th is a fear-of-mathematics joke. The sign the warrior’s carrying is legitimate algebra, at least so far as it goes. The right-hand side of the equation gets cut off. In time, it would get to the conclusion that x equals –19/2, or -9.5.

A Leap Day 2016 Mathematics A To Z: Quaternion

I’ve got another request from Gaurish today. And it’s a word I had been thinking to do anyway. When one looks for mathematical terms starting with ‘q’ this is one that stands out. I’m a little surprised I didn’t do it for last summer’s A To Z. But here it is at last:

Quaternion.

I remember the seizing of my imagination the summer I learned imaginary numbers. If we could define a number i, so that i-squared equalled negative 1, and work out arithmetic which made sense out of that, why not do it again? Complex-valued numbers are great. Why not something more? Maybe we could also have some other non-real number. I reached deep into my imagination and picked j as its name. It could be something else. Maybe the logarithm of -1. Maybe the square root of i. Maybe something else. And maybe we could build arithmetic with a whole second other non-real number.

My hopes of this brilliant idea petered out over the summer. It’s easy to imagine a super-complex number, something that’s “1 + 2i + 3j”. And it’s easy to work out adding two super-complex numbers like this together. But multiplying them together? What should i times j be? I couldn’t solve the problem. Also I learned that we didn’t need another number to be the logarithm of -1. It would be π times i. (Or some other numbers. There’s some surprising stuff in logarithms of negative or of complex-valued numbers.) We also don’t need something special to be the square root of i, either. $\frac{1}{2}\sqrt{2} + \frac{1}{2}\sqrt{2}\imath$ will do. (So will another number.) So I shelved the project.

Even if I hadn’t given up, I wouldn’t have invented something. Not along those lines. Finer minds had done the same work and had found a way to do it. The most famous of these is the quaternions. It has a famous discovery. Sir William Rowan Hamilton — the namesake of “Hamiltonian mechanics”, so you already know what a fantastic mind he was — had a flash of insight that’s come down in the folklore and romance of mathematical history. He had the idea on the 16th of October, 1843, while walking with his wife along the Royal Canal, in Dublin, Ireland. While walking across the bridge he saw what was missing. It seems he lacked pencil and paper. He carved it into the bridge:

$i^2 = j^2 = k^2 = ijk = -1$

The bridge now has a plaque commemorating the moment. You can’t make a sensible system with two non-real numbers. But three? Three works.

And they are a mysterious three! i, j, and k are somehow not the same number. But each of them, multiplied by themselves, gives us -1. And the product of the three is -1. They are even more mysterious. To work sensibly, i times j can’t be the same thing as j times i. Instead, i times j equals minus j times i. And j times k equals minus k times j. And k times i equals minus i times k. We must give up commutivity, the idea that the order in which we multiply things doesn’t matter.

But if we’re willing to accept that the order matters, then quaternions are well-behaved things. We can add and subtract them just as we would think to do if we didn’t know they were strange constructs. If we keep the funny rules about the products of i and j and k straight, then we can multiply them as easily as we multiply polynomials together. We can even divide them. We can do all the things we do with real numbers, only with these odd sets of four real numbers.

The way they look, that pattern of 1 + 2i + 3j + 4k, makes them look a lot like vectors. And we can use them like vectors pointing to stuff in three-dimensional space. It’s not quite a comfortable fit, though. That plain old real number at the start of things seems like it ought to signify something, but it doesn’t. In practice, it doesn’t give us anything that regular old vectors don’t. And vectors allow us to ponder not just three- or maybe four-dimensional spaces, but as many as we need. You might wonder why we need more than four dimensions, even allowing for time. It’s because if we want to track a lot of interacting things, it’s surprisingly useful to put them all into one big vector in a very high-dimension space. It’s hard to draw, but the mathematics is nice. Hamiltonian mechanics, particularly, almost beg for it.

That’s not to call them useless, or even a niche interest. They do some things fantastically well. One of them is rotations. We can represent rotating a point around an arbitrary axis by an arbitrary angle as the multiplication of quaternions. There are many ways to calculate rotations. But if we need to do three-dimensional rotations this is a great one because it’s easy to understand and easier to program. And as you’d imagine, being able to calculate what rotations do is useful in all sorts of applications.

They’ve got good uses in number theory too, as they correspond well to the different ways to solve problems, often polynomials. They’re also popular in group theory. They might be the simplest rings that work like arithmetic but that don’t commute. So they can serve as ways to learn properties of more exotic ring structures.

Knowing of these marvelous exotic creatures of the deep mathematics your imagination might be fired. Can we do this again? Can we make something with, say, four unreal numbers? No, no we can’t. Four won’t work. Nor will five. If we keep going, though, we do hit upon success with seven unreal numbers.

This is a set called the octonions. Hamilton had barely worked out the scheme for quaternions when John T Graves, a friend of his at least up through the 16th of December, 1843, wrote of this new scheme. (Graves didn’t publish before Arthur Cayley did. Cayley’s one of those unspeakably prolific 19th century mathematicians. He has at least 967 papers to his credit. And he was a lawyer doing mathematics on the side for about 250 of those papers. This depresses every mathematician who ponders it these days.)

But where quaternions are peculiar, octonions are really peculiar. Let me call a couple quaternions p, q, and r. p times q might not be the same thing as q times r. But p times the product of q and r will be the same thing as the product of p and q itself times r. This we call associativity. Octonions don’t have that. Let me call a couple quaternions s, t, and u. s times the product of t times u may be either positive or negative the product of s and t times u. (It depends.)

Octonions have some neat mathematical properties. But I don’t know of any general uses for them that are as catchy as understanding rotations. Not rotations in the three-dimensional world, anyway.

Yes, yes, we can go farther still. There’s a construct called “sedenions”, which have fifteen non-real numbers on them. That’s 16 terms in each number. Where octonions are peculiar, sedenions are really peculiar. They work even less like regular old numbers than octonions do. With octonions, at least, when you multiply s by the product of s and t, you get the same number as you would multiplying s by s and then multiplying that by t. Sedenions don’t even offer that shred of normality. Besides being a way to learn about abstract algebra structures I don’t know what they’re used for.

I also don’t know of further exotic terms along this line. It would seem to fit a pattern if there’s some 32-term construct that we can define something like multiplication for. But it would presumably be even less like regular multiplication than sedenion multiplication is. If you want to fiddle about with that please do enjoy yourself. I’d be interested to hear if you turn up anything, but I don’t expect it’ll revolutionize the way I look at numbers. Sorry. But the discovery might be the fun part anyway.

Reading the Comics, July 1, 2012

This will be a hastily-written installment since I married just this weekend and have other things occupying me. But there’s still comics mentioning math subjects so let me summarize them for you. The first since my last collection of these, on the 13th of June, came on the 15th, with Dave Whamond’s Reality Check, which goes into one of the minor linguistic quirks that bothers me: the claim that one can’t give “110 percent,” since 100 percent is all there is. I don’t object to phrases like “110 percent”, though, since it seems to me the baseline, the 100 percent, must be to some standard reference performance. For example, the Space Shuttle Main Engines routinely operated at around 104 percent, not because they were exceeding their theoretical limits, but because the original design thrust was found to be not quite enough, and the engines were redesigned to deliver more thrust, and it would have been far too confusing to rewrite all the documentation so that the new design thrust was the new 100 percent. Instead 100 percent was the design capacity of an engine which never flew but which existed in paper form. So I’m forgiving of “110 percent” constructions, is the important thing to me.

Reading The Comics, May 20, 2012

Since I suspect that the comics roundup posts are the most popular ones I post, I’m very glad to see there was a bumper crop of strips among the ones I read regularly (from King Features Syndicate and from gocomics.com) this past week. Some of those were from cancelled strips in perpetual reruns, but that’s fine, I think: there aren’t any particular limits on how big an electronic comics page one can have, after all, and while it’s possible to read a short-lived strip long enough that you see all its entries, it takes a couple go-rounds to actually have them all memorized.

The first entry, and one from one of these cancelled strips, comes from Mark O’Hare’s Citizen Dog, a charmer of a comic set in a world-plus-talking-animals strip. In this case Fergus has taken the place of Maggie, a girl who’s not quite ready to come back from summer vacation. It’s also the sort of series of questions that it feels like come at the start of any class where a homework assignment’s due.

How Many Numbers Have We Named?

I want to talk about some numbers which have names, and to argue that surprisingly few of numbers do. To make that argument it would be useful to say what numbers I think have names, and which ones haven’t; perhaps if I say enough I will find out.

For example, “one” is certainly a name of a number. So are “two” and “three” and so on, and going up to “twenty”, and going down to “zero”. But is “twenty-one” the name of a number, or just a label for the number described by the formula “take the number called twenty and add to it the number called one”?

It feels to me more like a label. I note for support the former London-dialect preference for writing such numbers as one-and-twenty, two-and-twenty, and so on, a construction still remembered in Charles Dickens, in nursery rhymes about blackbirds baked in pies, in poetry about the ways of constructing tribal lays correctly. It tells you how to calculate the number based on a few named numbers and some operations.

None of these are negative numbers. I can’t think of a properly named negative number, just ones we specify by prepending “minus” or “negative” to the label given a positive number. But negative numbers are fairly new things, a concept we have found comfortable for only a few centuries. Perhaps we will find something that simply must be named.

That tips my attitude (for today) about these names, that I admit “thirty” and “forty” and so up to a “hundred” as names. After that we return to what feel like formulas: a hundred and one, a hundred and ten, two hundred and fifty. We name a number, to say how many hundreds there are, and then whatever is left over. In ruling “thirty” in as a name and “three hundred” out I am being inconsistent; fortunately, I am speaking of peculiarities of the English language, so no one will notice. My dictionary notes the “-ty” suffix, going back to old English, means “groups of ten”. This makes “thirty” just “three tens”, stuffed down a little, yet somehow I think of “thirty” as different from “three hundred”, possibly because the latter does not appear in my dictionary. Somehow the impression formed in my mind before I thought to look.
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