The next piece in this set tour is a hybrid. It mixes properties of the last two sets. And I’ll own up now that while it’s a set that gets used a lot, it’s one that gets used a lot in just some corners of mathematics. It’s got a bit of that “Internet fame”. In particular circles it’s well-known; venture outside those circles even a little, and it’s not. But it leads us into other, useful places.

## **C**^{n}

**C** here is the set of complex-valued numbers. We may have feared them once, but now they’re friends, or at least something we can work peacefully with. *n* here is some counting number, just as it is with **R**^{n}. *n* could be one or two or forty or a hundred billion. It’ll be whatever fits the problem we’re doing, if we need to pin down its value at all.

The reference to **R**^{n}, another friend, probably tipped you off to the rest. The items in **C**^{n} are n-tuples, ordered sets of some number n of numbers. Each of those numbers is itself a complex-valued number, something from **C**. **C**^{n} gets typeset in bold, and often with that extra vertical stroke on the left side of the C arc. It’s handwritten that way, too.

As with **R**^{n} we can add together things in **C**^{n}. Suppose that we are in **C**^{2} so that I don’t have to type too much. Suppose the first number is (2 + i, -3 – 3*i) and the second number is (6 – 2*i, 2 + 9*i). There could be fractions or irrational numbers in the real and imaginary components, but I don’t want to type that much. The work is the same. Anyway, the sum will be another number in **C**^{n}. The first term in that sum will be the sum of the first term in the first number, 2 + i, and the first term in the second number, 6 – 2*i. That in turn will be the sum of the real and of the imaginary components, so, 2 + 6 + i – 2*i, or 8 – i all told. The second term of the sum will be the second term of the first number, -3 – 3*i, and the second term of the second number, 2 + 9*i, which will be -3 – 3*i + 2 + 9*i or, all told, -1 + 6*i. The sum is the n-tuple (8 – i, -1 + 6*i).

And also as with **R**^{n} there really isn’t multiplying of one term of **C**^{n} by another. Generally, we can’t do this in any useful way. We *can* multiply something in **C**^{n} by a scalar, a single real — or, why not, complex-valued — number, though.

So let’s start out with (8 – i, -1 + 6*i), a number in **C**^{2}. And then pick a scalar, say, 2 + 2*i. It doesn’t have to be complex-valued, but, why not? The product of this scalar and this term will be another number in **C**^{2}. Its first term will the scalar, 2 + 2*i, multiplied by the first term in it, 8 – i. That’s (2 + 2*i) * (8 – i), or 2*8 – 2*i + 16*i – 2*i*i, or 2*8 – 2*i + 16*i + 2, or 18 + 14*i. And then its second term will be the scalar 2 + 2*i multiplied by the second term, -1 + 6*i. That’s (2 + 2*i)*(-1 + 6*i), or 2*(-1) + 2*6*i -2*i + 2*6*i*i. And that’s -2 + 12*i – 2*i -12, or -14 + 10*i. So the product is (18 + 14*i, -14 + 10*i).

So as with **R**^{n}, **C**^{n} creates a “vector space”. These spaces are useful in complex analysis. They’re also useful in the study of affine geometry, a corner of geometry that I’m sad to admit falls outside what I studied. I have tried reading up on them on my own, and I run aground each time. I understand the basic principles but never quite grasp why they are interesting. That’s my own failing, of course, and I’d be glad for a pointer that explained in ways I understood why they’re so neat.

I do understand some of what’s neat about them: affine geometry tells us what we can know about shapes without using the concept of “distance”. When you discover that we *can* know anything about shapes without the idea of “distance” your imagination should be fired. Mine is, too. I just haven’t followed from that to feel comfortable with the terminology and symbols of the field.

You could, if you like, think of **C**^{n} as being a specially-delineated version of **R**^{2*n}. This is just as you can see a complex number as an ordered pair of real numbers. But sometimes information is usefully thought of as a single, complex-valued number. And there is a value in introducing the idea of ordered sets of things that are not real numbers. We will see the concept again.

Also, the *heck* did I write an 800-word essay about the family of sets of complex-valued n-tuples and have Hemingway Editor judge it to be at the “Grade 3” reading level? I rarely get down to “Grade 6” when I do a Reading the Comics post explaining how Andertoons did a snarky-word-problem-answers panel. That’s got to be a temporary glitch.

Can we give meaning to the scalar or dot product in C^n ?

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Oh, absolutely. We do have to start making some tough choices, though. The most obvious way of defining a dot product means that it’s possible to find something nonzero whose dot product with itself is a negative number, or even zero. (For the non mathematician: the dot product of a thing with itself is often the easiest way to define the length of something, so you can see why that’s a weird effect.)

But you can work around that, at the cost of other weirdness. For example, you can lose symmetry — the dot product of thing one with thing two is no longer (necessarily) equal to the dot product of thing two with thing one. (Again for the non mathematician, that’s something we expect

somuch by default it’s a shock to see it not happen.)That workaround — it’s done by multiplying terms in the first thing from C^n by the complex conjugate of the corresponding terms in the second thing, and adding up these sums — is probably the most useful. You can use it to define the angle between terms in C^n and that’s so hard to dispense with.

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Thanks, Joseph. We never got this far !

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