My Little 2021 Mathematics A-to-Z: Subtraction


Iva Sallay was once again a kind friend to my writing efforts here. Sallay, who runs the Find the Factors recreational mathematics puzzle site, saw a topic gives a compelling theme to this year’s A-to-Z.

Subtraction.

Subtraction is the inverse of addition.

So thanks for reading along as the Little 2021 Mathematics A-to-Z enters its final stage. Next week I hope to be back with something for my third letter ‘A’ of the sequence.

All right, I can be a little more clear. By the inverse I mean subtraction is the name the name we give to adding the additive inverse of something. It’s what lets addition be a group action. That is, we write a - b to mean we find whatever number, added to b, gives us 0. Then we add that to a. We do this pretty often, so it’s convenient to have a name for it. The word “subtraction” appears in English from about 1400. It grew from the Latin for “taking away”. By about 1425 the word has its mathematical meaning. I imagine this wasn’t too radical a linguistic evolution

All right, so some other thoughts. What’s so interesting about subtraction that it’s worth a name? We don’t have a particular word for reversing, say, a permutation. But don’t go very far in school not thinking about inverting an addition. Must come down to subtraction’s practical use in finding differences between things. Often in figuring out change. Debts at least. Nobody needs the inverse of a permutation unless they’re putting a deck of cards back in order.

Subtraction has other roles, though. Not so much in mathematics, but in teaching us how to learn about mathematics. For example, subtraction gives us a good reason to notice zero. Zero, the additive identity, is implicit to addition. But if you’re learning addition, and you think of it as “put these two piles of things together into one larger pile”? What good does an empty pile do you there? It’s easy to not notice there’s a concept there. But subtraction, taking stuff away from a pile? You can imagine taking everything away, and wanting a word for that. This isn’t the only way to notice zero is worth some attention. It’s a good way, though.

There’s more, though. Learning subtraction teaches us limits of what we can do, mathematically. We can add 3 to 7 or, if it’s more convenient, 7 to 3. But we learn from the start that while we can subtract 3 from 7, there’s no subtracting 7 from 3. This is true when we’re learning arithmetic and numbers are all positive. Some time later we ask, what happens if we go ahead and do this anyway? And figure out a number that makes sense as the answer to “what do you get subtracting 7 from 3”? This introduces us to the negative numbers. It’s a richer idea of what it is to have numbers. We can start to see addition and subtraction as expressions of the same operation.

Linus: 'Lucy, how much is six from four?' Lucy: 'Six from four?! You can't subtract six from four ... you can't subtract a bigger number from a smaller number.' Linus: 'YOU CAN IF YOU'RE STUPID!'
Charles Schulz’s Peanuts for the 27th of August, 1957. The amazing thing is you can if you’re smart, too. We can ask whether it’s good teaching to start instructions with something that’s not true, and then revealing what’s not true about it. My hunch is there is, because this provides the lesson that, even for something as “objective” as mathematics, the way we construct things is a convention. That we can change our tools as we want to do new things.

But we also notice they’re not quite the same. As mentioned, addition can be done in any order. If I need to do 7 + 4 + 3 + 6 I can decide I’d rather do 4 + 6 + 7 + 3 and make that 10 + 10 before getting to 20. This all simplifies my calculating. If I need to do 7 – 4 – 3 – 6 I get into a lot of trouble if I simplify my work by writing 4 – 6 – 7 – 3 instead. Even if I decide I’d rather take the 3 – 6 and turn that into a negative 3 first, I’ve made a mess of things.

The first property this teaches us to notice we call “commutativity”. Most mathematical operations don’t have that. But a lot of the ones we find useful do. The second property this points out is “associativity”, which more of the operations we find useful have. It’s not essential that someone learning how to calculate know this is a way to categorize mathematics operations. (I’ve read that before the New Math educational reforms of the 1960s, American elementary school mathematics textbooks never mentioned commutativity or associativity.) But I suspect it is essential that someone learning mathematics learn the things you can do come in families.

So let me mention division, the inverse of multiplication. (And that my chosen theme won’t let me get to in sequence.) Like subtraction, division refuses to be commutative or associative. Subtraction prompts us to treat the negative numbers as something useful. In parallel, division prompts us to accept fractions as numbers. (We accepted fractions as numbers long before we accepted negative numbers, mind. Anyone with a pie and three friends has an interest in “one-quarter” that they may not have with “negative four”.) When we start learning about numbers raised to powers, or exponentials, we have questions ready to ask. How do the operations behave? Do they encourage us to find other kinds of number?

And we also think of how to patch up subtraction’s problems. If we want subtraction to be a kind of addition, we have to get precise about what that little subtraction sign means. What we’ve settled on is that a - b is shorthand for a + (-b) , where -b is the additive inverse of b .

Once we do that all subtraction’s problems with commutativity and associativity go away. 7 – 4 – 3 – 6 becomes 7 + (-4) + (-3) + (-6), and that we can shuffle around however convenient. Say, to 7 + (-3) + (-4) + (-6), then to 7 + (-3) + (-10), then to 4 + (-10), and so -6. Thus do we domesticate a useful, wild operation like subtraction.

Any individual subtraction has one right answer. There are many ways to get there, though. I had learned, for example, to do a problem such as 738 minus 451 by subtracting one column of numbers at a time. Right to left, so, subtracting 8 minus 1, and then 3 minus 5, and after the borrowing then 6 minus 4. I remember several elementary school textbooks explaining borrowing as unwrapping rolls of dimes. It was a model well-suited to me.

We don’t need to, though. We can go from the left to the right, doing 7 minus 4 first and 8 minus 1 last. We can go through and figure out all the possible carries before doing any work. There’s a slick method called partial differences which skips all the carrying. But it demands writing out several more intermediate terms. This uses more paper, but if there isn’t a paper shortage, so what?

There are more ways to calculate. If we turn things over to a computer, we’re likely to do subtraction using a complements technique. When I say computer you likely think electronic computer, or did right up to the adjective there. But mechanical computers were a thing too. Blaise Pascal’s computing device of the 1650s used nines’ complements to subtract on the gears that did addition. Explaining the trick would take me farther afield than I want to go now. But, you know how, like, 6 plus 3 is 9? So you can turn a subtraction of 6 into an addition of 3. Or a subtraction of 3 into an addition of 6. Plus some bookkeeping.

A digital computer is likely to use ones’ complements, representing every number as a string of 0’s and 1’s. This has great speed advantages. The complement of 0 is 1 and vice-versa, and it’s very quick for a computer to swap between 0 and 1. Subtraction by complements is different and, to my eye, takes more steps. But they might be steps you do better.

One more thought subtraction gives us, though. In a previous paragraph I wrote out 7 – 4, and also wrote 7 + (-4). We use the symbol – for two things. Do those two uses of – mean the same thing? You may think I’m being fussy here. After all, the value of -4 is the same as the value of 0 – 4. And even a fussy mathematician says whichever of “minus four” and “negative four” better fits the meter of the sentence. But our friends in the philosophy department would agree this is a fair question. Are we collapsing two related ideas together by using the same symbol for them?

My inclination is to say that the – of -4 is different from the – in 0 – 4, though. The – in -4 is a unary operation: it means “give me the inverse of the number on the right”. The – in 0 – 4 is a binary operation: it means “subtract the number on the right from the number on the left”. So I would say these are different things sharing a symbol. Unfortunately our friends in the philosophy department can’t answer the question for us. The university laid them off four years ago, part of society’s realignment away from questions like “how can we recognize when a thing is true?” and towards “how can we teach proto-laborers to use Excel macros?”. We have to use subtraction to expand our thinking on our own.

My Little 2021 Mathematics A-to-Z: Inverse


I owe Iva Sallay thanks for the suggestion of today’s topic. Sallay is a longtime friend of my blog here. And runs the Find the Factors recreational mathematics puzzle site. If you haven’t been following, or haven’t visited before, this is a fun week to step in again. The puzzles this week include (American) Thanksgiving-themed pictures.

Inverse.

When we visit the museum made of a visual artist’s studio we often admire the tools. The surviving pencils and crayons, pens, brushes and such. We don’t often notice the eraser, the correction tape, the unused white-out, or the pages cut into scraps to cover up errors. To do something is to want to undo it. This is as true for the mathematics of a circle as it is for the drawing of one.

If not to undo something, we do often want to know where something comes from. A classic paper asks can one hear the shape of a drum? You hear a sound. Can you say what made that sound? Fine, dismiss the drum shape as idle curiosity. The same question applies to any sensory data. If our hand feels cooler here, where is the insulation of the building damaged? If we have this electrocardiogram reading, what can we say about the action of the heart producing that? If we see the banks of a river, what can we know about how the river floods?

And this is the point, and purpose, of inverses. We can understand them as finding the causes of what we observe.

The first inverse we meet is usually the inverse function. It’s introduced as a way to undo what a function does. That’s an odd introduction, if you’re comfortable with what a function is. A function is a mathematical construct. It’s two sets — a domain and a range — and a rule that links elements in the domain to the range. To “undo” a function is like “undoing” a rectangle. But a function has a compelling “physical” interpretation. It’s routine to introduce functions as machines that take some numbers in and give numbers out. We think of them as ways to transform the domain into the range. In functional analysis get to thinking of domains as the most perfect putty. We expect functions to stretch and rotate and compress and slide along as though they were drawing a Betty Boop cartoon.

So we’re trained to speak of a function as a verb, acting on pieces of the domain. An element or point, or a region, or the whole domain. We think the function “maps”, or “takes”, or “transforms” this into its image in the range. And if we can turn one thing into another, surely we can turn it back.

Some things it’s obvious we can turn back. Suppose our function adds 2 to whatever we give it. We can get the original back by subtracting 2. If the function subtracts 32 and divides by 1.8, we can reverse it by multiplying by 1.8 and adding 32. If the function takes the reciprocal, we can take the reciprocal again. We have a bit of a problem if we started out taking the reciprocal of 0, but who would want to do such a thing anyway? If the function squares a number, we can undo that by taking the square root. Unless we started from a negative number. Then we have trouble.

The trouble is not every function has an inverse. Which we could have realized by thinking how to undo “multiply by zero”. To be a well-defined function, the rule part has to match elements in the domain to exactly one element in the range. This makes the function, in the impenetrable jargon of the mathematician, a “one-to-one function”. Or you can describe it with the more intuitive label of “bijective”.

But there’s no reason more than one thing in the domain can’t match to the same thing in the range. If I know the cosine of my angle is \frac{1}{2}, my angle might be 30 degrees. Or -30 degrees. Or 390 degrees. Or 330 degrees. You may protest there’s no difference between a 30 degree and a 390 degree angle. I agree those angles point in the same direction. But a gear rotated 390 degrees has done something that a gear rotated 30 degrees hasn’t. If all I know is where the dot I’ve put on the gear is, how can I know how much it’s rotated?

So what we do is shift from the actual cosine into one branch of the cosine. By restricting the domain we can create a function that has the same rule as the one we want, but that’s also one-to-one and so has an inverse. What restriction to use? That depends on what you want. But mathematicians have some that come up so often they might as well be defaults. So the square root is the inverse of the square of nonnegative numbers. The inverse Cosine is the inverse of the cosine of angles from 0 to 180 degrees. The inverse Sine is the inverse of the sine of angles from -90 to 90 degrees. The capital letters are convention to say we’re doing this. If we want a different range, we write out that we’re looking for an inverse cosine from -180 to 0 degrees or whatever. (Yes, the mathematician will default to using radians, rather than degrees, for angles. That’s a different essay.) It’s an imperfect solution, but it often works well enough.

The trouble we had with cosines, and functions, continues through all inverses. There are almost always alternate causes. Many shapes of drums sound alike. Take two metal bars. Heat both with a blowtorch, one on the end and one in the center. Not to the point of melting, only to the point of being too hot to touch. Let them cool in insulated boxes for a couple weeks. There’ll be no measurement you can do on the remaining heat that tells you which one was heated on the end and which the center. That’s not because your thermometers are no good or the flow of heat is not deterministic or anything. It’s that both starting cases settle to the same end. So here there is no usable inverse.

This is not to call inverses futile. We can look for what we expect to find useful. We are inclined to find inverses of the cosine between 0 and 180 degrees, even though 4140 through 4320 degrees is as legitimate. We may not know what is wrong with a heart, but have some idea what a heart could do and still beat. And there’s a famous example in 19th-century astronomy. After the discovery of Uranus came the discovery it did not move right. For a while it moved across the sky too fast for its distance from the sun. Then it started moving too slow. The obvious supposition was that there was another, not-yet-seen, planet, affecting its orbit.

The trouble is finding it. Calculating the orbit from what data they had required solving equations with 13 unknown quantities. John Couch Adams and Urbain Le Verrier attempted this anyway, making suppositions about what they could not measure. They made great suppositions. Le Verrier made the better calculations, and persuaded an astronomer (Johann Gottfried Galle, assisted by Heinrich Louis d’Arrest) to go look. Took about an hour of looking. They also made lucky suppositions. Both, for example, supposed the trans-Uranian planet would obey “Bode’s Law”, a seeming pattern in the size of planetary radiuses. The actual Neptune does not. It was near enough in the sky to where the calculated planet would be, though. The world is vaster than our imaginations.

That there are many ways to draw Betty Boop does not mean there’s nothing to learn about how this drawing was done. And so we keep having inverses as a vibrant field of mathematics.


Next week I hope to cover the letter ‘C’ and don’t think I’m not worried about what that ‘C’ will be. This week’s essay, and all the essays for the Little Mathematics A-to-Z, should be at this link. And all of this year’s essays, and all the A-to-Z essays from past years, should be at this link. Thank you for reading.

The End 2016 Mathematics A To Z: Principal


Functions. They’re at the center of so much mathematics. They have three pieces: a domain, a range, and a rule. The one thing functions absolutely must do is match stuff in the domain to one and only one thing in the range. So this is where it gets tricky.

Principal.

Thing with this one-and-only-one thing in the range is it’s not always practical. Sometimes it only makes sense to allow for something in the domain to match several things in the range. For example, suppose we have the domain of positive numbers. And we want a function that gives us the numbers which, squared, are whatever the original function was. For any positive real number there’s two numbers that do that. 4 should match to both +2 and -2.

You might ask why I want a function that tells me the numbers which, squared, equal something. I ask back, what business is that of yours? I want a function that does this and shouldn’t that be enough? We’re getting off to a bad start here. I’m sorry; I’ve been running ragged the last few days. I blame the flat tire on my car.

Anyway. I’d want something like that function because I’m looking for what state of things makes some other thing true. This turns up often in “inverse problems”, problems in which we know what some measurement is and want to know what caused the measurement. We do that sort of problem all the time.

We can handle these multi-valued functions. Of course we can. Mathematicians are as good at loopholes as anyone else is. Formally we declare that the range isn’t the real numbers but rather sets of real numbers. My what-number-squared function then matches ‘4’ in the domain to the set of numbers ‘+2 and -2’. The set has several things in it, but there’s just the one set. Clever, huh?

This sort of thing turns up a lot. There’s two numbers that, squared, give us any real number (except zero). There’s three numbers that, squared, give us any real number (again except zero). Polynomials might have a whole bunch of numbers that make some equation true. Trig functions are worse. The tangent of 45 degrees equals 1. So is the tangent of 225 degrees. Also 405 degrees. Also -45 degrees. Also -585 degrees. OK, a mathematician would use radians instead of degrees, but that just changes what the numbers are. Not that there’s infinitely many of them.

It’s nice to have options. We don’t always want options. Sometimes we just want one blasted simple answer to things. It’s coded into the language. We say “the square root of four”. We speak of “the arctangent of 1”, which is to say, “the angle with tangent of 1”. We only say “all square roots of four” if we’re making a point about overlooking options.

If we’ve got a set of things, then we can pick out one of them. This is obvious, which means it is so very hard to prove. We just have to assume we can. Go ahead; assume we can. Our pick of the one thing out of this set is the “principal”. It’s not any more inherently right than the other possibilities. It’s just the one we choose to grab first.

So. The principal square root of four is positive two. The principal arctangent of 1 is 45 degrees, or in the dialect of mathematicians π divided by four. We pick these values over other possibilities because they’re nice. What makes them nice? Well, they’re nice. Um. Most of their numbers aren’t that big. They use positive numbers if we have a choice in the matter. Deep down we still suspect negative numbers of being up to something.

If nobody says otherwise then the principal square root is the positive one, or the one with a positive number in front of the imaginary part. If nobody says otherwise the principal arcsine is between -90 and +90 degrees (-π/2 and π/2). The principal arccosine is between 0 and 180 degrees (0 and π), unless someone says otherwise. The principal arctangent is … between -90 and 90 degrees, unless it’s between 0 and 180 degrees. You can count on the 0 to 90 part. Use your best judgement and roll with whatever develops for the other half of the range there. There’s not one answer that’s right for every possible case. The point of a principal value is to pick out one answer that’s usually a good starting point.

When you stare at what it means to be a function you realize that there’s a difference between the original function and the one that returns the principal value. The original function has a range that’s “sets of values”. The principal-value version has a range that’s just one value. If you’re being kind to your audience you make some note of that. Usually we note this by capitalizing the start of the function: “arcsin z” gives way to “Arcsin z”. “Log z” would be the principal-value version of “log z”. When you start pondering logarithms for negative numbers or for complex-valued numbers you get multiple values. It’s the same way that the arcsine function does.

And it’s good to warn your audience which principal value you mean, especially for the arc-trigonometric-functions or logarithms. (I’ve never seen someone break the square root convention.) The principal value is about picking the most obvious and easy-to-work-with value out of a set of them. It’s just impossible to get everyone to agree on what the obvious is.

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