I don’t know why that one caught my eye, but here I go.
You’ve probably seen the number line before, a straight line from left to right (or right to left, if you like) with a point on the line for every real number. A real number, before you ask, is just that: real. You can see it in the world. If I point to a finger on my hand and ask, “how many of these do I have?”, the answer is a real number. So is the answer to “how tall am I?”, and the answer to “How much money do I have?” The answer to that last question, notice, might be less than nothing but it would still be real for all that.
Alright, what if you have a number line on a piece of paper, and then turn the paper around by half of half a turn, so the number line has to run from top to bottom instead of left to right? It’s still a number line, of course. But now you can draw another number line from left to right, so that it will go over the first line. Then you have not one line but two.
What if you next put a point on the paper? Because there is not one number line but two, the point can mean not just one real number but two. You can read off the first from the left-to-right line, and then a second from the top-to-bottom line. And here is a funny thing: since you still have just one point on the paper, you still have one number — at the very same time that you have two!
I recognize that this may confuse. What’s the deal? The thing to see is that the one-number-that-is-really-two is a new kind of number. It is not a real number! It’s a different kind of number which I’ll call a complete number. (That’s not the name other people use, but it is not much different and will have to do.) So there is no problem here, because a complete number is a different kind of number to a real number. A complete number is like a pair of jeans with a left leg and a right leg; each leg is a real number, and the two together make up a pair.
Why go to all this trouble for a complete number that isn’t even real? Well, sometimes when you ask a question about a real number, the answer to the question is a complete number, even if you might expect the answer to be a real number. You can get angry and shout that you don’t want an answer that’s complete, and that you only want to work with a number if it’s real, but then you’ll find many a question you just can’t answer. But you can answer them all if you’re cool with an answer that’s complete!
For what it’s worth, I dislike the term “real number” precisely because it suggests that there’s something particularly real about them. Real numbers have a consistent and unambiguous mathematical definition; so do complex numbers. Real numbers show up in the real world; so do complex numbers. If I were to tell someone about real numbers, I would immediately mention that there’s nothing that makes them any more real or fake than any other kind of number.
Unrelatedly, my favorite mathematical definition (the one that I enjoy the most, not the one I think is actually best in any sense) is essentially the opposite of Up-Goer Five: it tries to explain a concept as thoroughly as possible using as few words as possible, even if that requires using very obscure words. That definition is:
The complex numbers are the algebraic closure of the completion of the field of fractions of the initial ring.
I thought I might get some pushback on taking the word “real” in “real number” literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.
We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result’s almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.
But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction’s real & imaginary parts.
In the macroscopic world it’s not easy to physically get at whatever complex numbers comprise a system’s state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.
As such I’m OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say “real numbers are just right there”, while the same isn’t true of complex numbers. This isn’t especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they’d get out of, say, “real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism”.)
As far as I know, the most visible way that complex numbers show up “in the real world” is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.
Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.
I don’t know why that one caught my eye, but here I go.
You’ve probably seen the number line before, a straight line from left to right (or right to left, if you like) with a point on the line for every real number. A real number, before you ask, is just that: real. You can see it in the world. If I point to a finger on my hand and ask, “how many of these do I have?”, the answer is a real number. So is the answer to “how tall am I?”, and the answer to “How much money do I have?” The answer to that last question, notice, might be less than nothing but it would still be real for all that.
Alright, what if you have a number line on a piece of paper, and then turn the paper around by half of half a turn, so the number line has to run from top to bottom instead of left to right? It’s still a number line, of course. But now you can draw another number line from left to right, so that it will go over the first line. Then you have not one line but two.
What if you next put a point on the paper? Because there is not one number line but two, the point can mean not just one real number but two. You can read off the first from the left-to-right line, and then a second from the top-to-bottom line. And here is a funny thing: since you still have just one point on the paper, you still have one number — at the very same time that you have two!
I recognize that this may confuse. What’s the deal? The thing to see is that the one-number-that-is-really-two is a new kind of number. It is not a real number! It’s a different kind of number which I’ll call a complete number. (That’s not the name other people use, but it is not much different and will have to do.) So there is no problem here, because a complete number is a different kind of number to a real number. A complete number is like a pair of jeans with a left leg and a right leg; each leg is a real number, and the two together make up a pair.
Why go to all this trouble for a complete number that isn’t even real? Well, sometimes when you ask a question about a real number, the answer to the question is a complete number, even if you might expect the answer to be a real number. You can get angry and shout that you don’t want an answer that’s complete, and that you only want to work with a number if it’s real, but then you’ll find many a question you just can’t answer. But you can answer them all if you’re cool with an answer that’s complete!
For what it’s worth, I dislike the term “real number” precisely because it suggests that there’s something particularly real about them. Real numbers have a consistent and unambiguous mathematical definition; so do complex numbers. Real numbers show up in the real world; so do complex numbers. If I were to tell someone about real numbers, I would immediately mention that there’s nothing that makes them any more real or fake than any other kind of number.
Unrelatedly, my favorite mathematical definition (the one that I enjoy the most, not the one I think is actually best in any sense) is essentially the opposite of Up-Goer Five: it tries to explain a concept as thoroughly as possible using as few words as possible, even if that requires using very obscure words. That definition is:
I thought I might get some pushback on taking the word “real” in “real number” literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.
We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result’s almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.
But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction’s real & imaginary parts.
In the macroscopic world it’s not easy to physically get at whatever complex numbers comprise a system’s state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.
As such I’m OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say “real numbers are just right there”, while the same isn’t true of complex numbers. This isn’t especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they’d get out of, say, “real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism”.)
As far as I know, the most visible way that complex numbers show up “in the real world” is as sine waves. Sine waves of a given frequency can be thought of as complex numbers. Adding together two sine waves corresponds to adding the corresponding complex numbers. Convoluting two sine waves corresponds to multiplying the corresponding complex numbers.
Since every analog signal can be thought of as a sum or integral of sine waves of different frequencies, an analog signal can be represented as a collection of complex numbers, one corresponding to the sinusoid at each frequency. This is what the Fourier transform is. Since convolution of analog signals corresponds to multiplication of their Fourier transforms, now a lot of the stuff we know about multiplication is applicable to convolution as well.