Practical uses for the number i

Factual Questions
21

Two possible approaches.

First is that complex numbers close our common mathematical system. In other words it answers the question “what if?”
Consider this- you can’t subtract 5 from 2. But what if you could? You get -3. It turns out the concept ia useful.
You can’t divide 3 by 7. But what if you could? You would get 3/7. It turbs out the concept is useful.
Ditto for complex numbers. i opens up maths that we couldn’t previously do and is useful.

Second approach. Complex numbers describe rotations. That is probably more accessible than wave functions. Complex numbees can be used to describe a coordinate as a single number. That is pretty powerful and quite cool.
Related to both of these is the concept of a vector. It is one thing to say that an object is moved 3 metres, but it leaves you with no idea of its final position. Negative three metrs can be useful since it describes the return trip. But in this sustem you are limited to one dimension - a road or train track. Complex numbers give you a distance and direction. Or alternatively a distance in the x direction and a distance in the y direction.

To my warpe way if thinking, the first explanation settles the existance of i and is the more compelling. The second telks you why this is such a big deal. There are other ways of describing these things but complex numbers are efficient and elegant.

22

Another EE checking in. Basically, it all starts with Euler’s formula:

e^ix = cos x + i sin x

which describes the unit circle where the horizontal axis is the real number line and the vertical axis is the imaginary number line.

From the wiki: Euler’s formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation “our jewel” and "the most remarkable formula in mathematics

And Isaac Asimov had a pretty good essay about forty years ago on the subject. Paraphrased here, couldn’t find the original text.

Better Explained has a pretty good explanation which begins—as mentioned upthread—with really understanding negative numbers: A Visual, Intuitive Guide to Imaginary Numbers – BetterExplained

23

From Better Explained: “A Visual, Intuitive Guide to Imaginary Numbers.”

24

I’m actually going to be talking to him about polar form of complex numbers this week. I’ll bring up e^ix, but probably won’t have time to go into it in detail, and I’m not sure if there’s an easy way to make the relation clear. Some of the math may be beyond his pay grade (and mine). That better explained link is great, I will use it.

Thanks again.

25

I think it helps to realise that the concept of number is necessarily abstract.

Show me a five!
You either end up showing me a symbol for five - a mark on paper that represents an abstract concept. Or else you show me five discrete objects. (Fish in your case.) To which I pont out that you have shownme a set of fish and are yet to show me a five. You then explain at lngth that five is a property of this particular set of fish and ut desrcibes it in some way. I show you five on my stopwatch and ask you how the two fives are in any way related.
After some discussion we agree that fiveness is an abstract concept that is useful for describing the quantity of certain things. Similarly i is used for describing certain things. Just like five, it has never been seen in isolation from the thing being described. Nevertheless, like five, the concept is logical and valid and its application in describing certain things useful.

26

Resistance is a simple concept in DC electricity. However, when using AC, resistance isn’t enough to fully describe things and impedance, or complex resistance, is used. My contribution: similarly, elastic modulus is a material property used in static situations. But in dynamic conditions (similar to AC), dynamic modulus is used, which uses complex numbers, similar to impedance. Dynamic modulus - Wikipedia

But my understanding of why we use complex numbers (maybe someone can correct me) isn’t to express 2 things “as a single number” - because a complex number isn’t really a single number. It has real and imaginary parts. If you’re storing a complex number in computer memory, it needs as much space as 2 numbers. If we simply wanted to store 2 numbers, we could use a vector.

So imagine we don’t have complex numbers and use vectors. There are some common operations we perform on these vectors. Suppose we have vectors (a,b) and (c,d), and define a common operation # such as:
(a,b)#(c,d) = (ac-bd,ad+bc)

might be so common that it’s a standard vector operation, like dot or cross product, or the determinant of a matrix. But rather than define a new operation, we might notice that # can simply be imagined as multiplying (a+ib) and (c+id), with i² = -1.

So I think the use of complex numbers is to define a set of rules and operations that are used in several applications, to save us from defining them all the time. It just so happens that the number i behaves in this way that we want.

27

Indistinguishable will surely be along shortly, since this is a favorite topic of his, but really, imaginary numbers aren’t any harder to imagine than negative numbers. Think back to elementary school, when you were learning about the number line. Multiplying by a negative number means you just turn everything around in the opposite direction. Multiplying by i, then, just means doing something that, if you do it twice, results in turning everything around in the opposite direction. Does it make sense to have something that, if you do it twice, turns you around? Sure, that’s no problem! A left turn meets exactly what we need. Or a right turn, either one. And those correspond exactly to ±i.

28

I had an introductory math textbook in college that referred to these numbers as “the j-operator”–just switching the letter j for the i. (This also avoids confusion with the letter I which is used in formulas for electric current.)
The word “imaginary” was never used in the text, except for a small footnote about terminology.

29

And here I am!

But I’m sort of tired, after my first day of working for a living (it’s no way to live!), so I’ll just note that a number of people in this thread have already said many of the things I would want to say anyway (including Gauss), and then copy-and-paste some things I’ve written before:

Complex numbers describe the arithmetic of rotation. This is such an incredibly intuitive and ubiquitous (even mundane!) kind of reasoning that it’s a wonder the meme of complex numbers as lacking physical application ever got started in the first place. What’s a more archetypal physical application than describing the geometry of our universe?

What do I mean when I say complex numbers describe the arithmetic of rotation? Well, let’s illustrate by thinking about some more familiar systems of arithmetic and physical interpretations thereof first, and then see where we can go from there.

Let’s think about sticks. Sticks have lengths. And we can scale these lengths; we can make a stick twice as big, or three times as big, or half as big, or 5.8 times as big, and so on. And it’s in terms of these length ratios that we actually give our measurements; we say “John is 5.8 feet tall” to mean “John is 5.8 times as big as a ruler; i.e., if you scaled a ruler by a factor of 5.8, it’d be as large as John”. And this shows us how to interpret certain numbers as actually about real-world quantities, and life is good.

And we can interpret addition and multiplication within this framework as well: multiplication means “chain the scalings one after another”: 7 * 5 = 35 because making something 7 times as large, and then making the result 5 times as large has the net effect of making what you started with 35 times as large. Addition means “carry out both scalings, then place the one stick after the other and see where you end up”" 7 + 5 = 12 because something 7 times as large as a ruler laid end to end with something 5 times as large as a ruler ends up at the same place as something 12 times as large as a ruler. So life is really good. We know perfectly well what arithmetic means now.

But wait… we’re missing something. We haven’t accounted for negative numbers. It wouldn’t seem like it means something to scale by a negative factor, so how can we make sense of them? Well, as you are probably familiar, there is a natural convention to adopt. Instead of focusing solely on lengths, we’ll now look at what direction our sticks are pointing in as well; in addition to scaling sticks up or down in size, we’ll also talk about flipping them 180 degrees around to point the other way. So, for example, -1 will mean “Turn your stick 180 degrees”, and -5 will mean “Make your stick 5 times as big and turn it 180 degrees”. But we’ll interpret addition and multiplication exactly the same way as before: -7 * 5 = -35 because “Make it 7 times as large and turn it 180 degrees” followed by “Make it 5 times as large” has the same net effect as “Make it 35 times as large and turn it 180 degrees”. And -7 + 5 = -2 because if I make two copies of my ruler, one 7 times as large but turned around, and the other 5 times as large and unturned, and place the one after the other, the ending point’s location is the same as if I’d just made a copy of my ruler which was twice as large and turned around. So life is super. Looks like negative numbers can be given physical sense in the same way as well.

But, hell, once we’ve started talking about turning sticks, why limit ourselves to full half-circle turns? Why not look at quarter-turns, eight-turns, 23.4 degree turns, and so on?

Why not indeed. Once we toss these in, we get… the complex numbers. All that mysterious i means is “Make a 90 degree turn”. We still interpret addition and multiplication exactly the same way as before; multiplication is still “Do these in sequence” and addition is still “Do these in parallel, lay the results one after another, and see where you end up.” In particular, as far as multiplication goes, since “Turn your stick 90 degrees. Now turn it 90 degrees again.” has the same net effect as “Turn your stick a full 180 degrees”, we see that i * i = -1. That’s it; it’s extraordinarily simple. Life is fantastic. Complex numbers are every bit as physically useful as real numbers; it’s just that the complex numbers express scaling with arbitrary rotation, while real numbers are limited to scaling with half-turn-increment rotation. [And non-negative real numbers express scaling with no rotation at all.] [And integers express…, and natural numbers express… Different systems for different purposes, that’s all.]

30

He’ll never have i fish, or i apples, or i cousins, but then, he’ll never have -1 cousins either. He’ll never come in 0.5th place in a race, and he’ll never enter a lottery with probability 2 of winning. Different number systems for different purposes.

31

In copying and pasting, I forgot that this use of the terminology “real number” could be confusing. I should have put it in scare quotes everywhere, because it is jargon (for, as noted above, the sorts of numbers which describe scaling with half-turn-increment rotation), and terribly named jargon at that (the name reflecting only fossilized ignorance, nothing more). “real numbers” are no more real than any other kinds of numbers, so please don’t draw that inference from my using that awful name.

32

If I were teaching math at the beginning algebra level (an occasional fantasy in which I indulge), maybe I would have a take on this.

It would go along these lines: Numbers are conceptual thingies, so we can always play mental games with them. But they are also tools that we use for specific tasks: Describing various things about the real world.

And different kinds of numbers are useful for describing different sorts of real things that exist in the real world.

Whole number are useful for counting whole things.
Fractions are useful for measuring amounts of things that aren’t whole: Distances, weights, time intervals, areas, etc.

Note that negative numbers aren’t useful for the above tasks. You can’t have -5 apples in your hand. If Johnny has 5 apples, he can’t give 7 of them to Susie. You can’t have a basket with -5.37 bushels of wheat. This is why negative numbers are confusing to beginners: They seem abstract and nothing but abstract.

BUT: Here is my introductory lecture on negative numbers. And guess what, it has nothing to do (at first glance) with numbers less than zero!

Suppose an airplane is flying at 35000 feet. The pilot changes his altitude by 400 feet. Now how high is he flying? The astute student will ask: Did he go 400 feet UP or 400 feet DOWN? The number 400, by itself, doesn’t tell us enough. So we have to add a qualifier (UP or DOWN) to the number. Many similar example could be given: The temperature, at first 40 degrees, changed 10 degrees. Up or down? Countable or measurable quantities can have “before” and “after” values, and they can change. (This concept, that variables are variable should, I think, be taught more thoroughly very early in algebra. I would even include teaching the Δx notation and terminology.)

So numbers can have direction to them. Mathematicians have decided that it is useful and convenient to think of the direction as part of the number – The airplane changes its altitude by 400↑ feet or 400↓ feet. The numbers 400↑ and 400↓ are two distinctly different numbers. Note that this explanation says nothing about any numbers being less than zero; nothing about Johnny and his -5 apples or Farmer Bill and his -85.7 bushels.

Once we get that understood, this we can discuss the notation: We write 400↓ as -400 and call it “negative 400” – and then I would discuss other useful interpretations. Now we can discuss why 0 (zero) isn’t really always 0 (as in nothing). What we call zero may just be an arbitrary or convenient starting point, like 0 altitude (sea-level) or 0 degrees (temperature), which aren’t really at the “bottom” of their scales. Thus, we can have numbers even smaller. Thus, negative numbers less than zero. Right here, in the real world! It’s not just an obscure abstract concept!

Okay, teach them all that early in their Algebra I class. Now they have all the right basic concepts: That numbers are useful to describe things in the real world; and that different kinds of numbers can be useful (or not) for describing different kinds of things in the real world. This will leave their minds open for the evil day that we must teach them even more weird kinds of numbers to describe even more weird kinds of things that are found in the real world. Thus, when the time comes to introduce imaginary numbers, their minds are already primed for the idea.

All you need to add are some plausible (understandable to 9th graders) kinds of situations that can be described neatly with imaginary numbers. Here, I would make some vague remarks to the effect that they are useful in electronics problems. And I would mention that they can be useful as directed numbers in a two-dimensional plane, not just as ↑ and ↓ numbers. At this level of their education, I doubt that I could give any actual examples in detail.

But a key thought here is to introduce all the earlier concepts (like fractions, negative numbers, variables, etc.) in ways that prime them to be ready for new ideas to be added as we go. I’m not so sure that this is typically done very well in most Algebra classes.

Here’s a thought: Beginning algebra students are taught that variables like x or y, as seen in equations, represent specific numbers. They are NOT taught until much later (if at all, in Algebra classes) that variables represent quantities that change. If a problem involves “before” and “after” numbers, two different variables are used to represent them.

So, early on, challenge your students with this question: If variables represent specific numbers, why are they called “variables”? Let them think about that for a while.

And, early on, challenge them with this question: Why are the Real numbers called Real numbers? Are there also Unreal numbers? Let them wonder about that for a while too.

33

One thing that I like about complex numbers is that you can use them to prove interesting facts about real numbers - and by the time you reach the end of the proof, all the imaginary terms have disappeared.

For instance, if you take two numbers that are sums of squares and multiply them together, the product is also a sum of squares: that is, 5 = 1[sup]2[/sup] + 2[sup]2[/sup], 13 = 2[sup]2[/sup] + 3[sup]2[/sup], and 13 x 5 = 65 = 1[sup]2[/sup] + 8[sup]2[/sup], and so on. You can prove this algebraically, and the proof begins with the complex factorisation x[sup]2[/sup] +y[sup]2[/sup] = (x + iy)(x - iy), and ends with all imaginary terms disappearing.

34

Thanks again, the rotation concept is really the best way to go about this, I think. It really does make it tangible. I think I’ll mention imaginary number’s use in electronics, but I won’t stress it. Saying “well some people use it” doesn’t seem like the most inspiring explanation, though it would be good for him to know.

Honestly, part of the problem is that this kid likes being difficult for the sake of being difficult. (He’s like a very brainy Bart Simpson) He’ll be fine with the concept. But still it’s a fair question, even if it is brought on by the unfortunate term “imaginary number.”

35

OT but congrats! And if it’s not too personal, what kind of job do you have now?

36

If you prefer, think of a special kind of matrices of the form [a,b;-b,a] (which of course include rotations, those being the special case that a^2 + b^2 = 1. For example, see what happens when you square [0,1;-1,0]. Surely no one doubts the value of matrices.

37

Another really good use of the number i is to freak out algebra students, by asking them to calculate i[sup]i[/sup].
It really can be done, and doing so reveals much about the number and its significance.

http://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml

38

I don’t have much to add here, except to say that if you study real numbers, “imaginary” numbers are easier to grasp, more concrete, and have more real world uses than real numbers.

Just as an example, the probability (I’m not a mathematician, so my terminology is probably all wrong) of picking a rational number at random on the real number line is 0. If you throw a dart at the number line, you will never hit a rational number exactly. If you break a stick in half, it will never be exactly 0.5. Then there’s the fact that real numbers are uncountable. And you have to get into calculus and measure theory to understand simple things like “length” and “continuity”.

On the other hand, “imaginary” numbers are just rotation. If positive numbers are 0 degrees, then negative numbers are 180 degrees, and you use “complex” numbers of the form a +bi (or A*e^jx) for any other kind of rotation. It makes it easy to do math with waveforms that are out of phase. Considering any signal at all is just a bunch of sine waves with different amplitudes and phases added together, and you can see how “imaginary” numbers relate pretty closely to the world we live in.

So the choice of terminology is unfortunate. “Real” numbers are abstract and difficult, often with few real world applications, while “imaginary” numbers are fairly simple and down to earth, with many concrete everyday uses.

But I guess that’s just my opinion. I’m an engineer too, so maybe my brain is different.

39

Isn’t it pretty much essential for Apple’s inventory management, as well?

And to stand for the imagined value of having a bitten-apple logo on your gear?

40

Ah yes. “i[sup]i[/sup] has infinitely many values, all of which are real”. Just when you thought mathematics was safe again.