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.