A complex number is not a solution to a previously unsolvable equation. It's not a formal addition to our mathematical system, without meaning but with numerous application. It's not a square root of a negative number.
A complex number is a point.
And, of course, a complex number is all those other things too. But the entire story makes the most sense if a complex number is just a two-part number, a point in the plane. The fact that it can be all those other things is just a wonderful surprise.
I am guessing that you are already familiar with http://mathforum.org/mbower/johnandbetty/frame.htm
Yours is a lovely, direct statement and I intend to incorporate this in my upcoming discussion with my BC kids. I am working on tying together trig/DeMoivre/complex numbers/vectors in what I hope is a meaningful discussion.
If I say to my kids that complex numbers are just points, is it logical to try and backtrack to the days of the number line to remind them that every real number is just a point?
Is that, sort of, the point of this statement of yours? Pun, of course, intended.
The idea that we can think of (a,0) as real numbers on the number line is really one of the enormous conceptual hurdles in the whole complex numbers business. It's central to the idea that what we're doing isn't just playing any game -- we're playing the conservative extension game, where we try to build something new around a familiar core.
I first thought (induced by missing date above it) that the next blog entry "Why We're Addicted To Online Outrage" belongs to this entry, and you simply intend to provoke us. Isn't it good that there is no notion of political correctness in math? If you think you know what a mathematical object IS that is okay with me. But you have to explain quaternions to me next time!ReplyDelete
Well, it is sort of obnoxious of me to claim to know what complex numbers definitely are. But, hey, this is the internet where unusually strong opinions thrive.
I think of quaternions as points in 4-space, or just as 4-part numbers. To me, that fits well with thinking of complex numbers as just 2-part numbers. How do you think of them?
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I think of quaternions as the covering group of the group of linear similitudes of |R3.ReplyDelete
I meant to say "universal covering group". Multiplicatively.ReplyDelete
This does lead to the definition of a numeral as "just a mark on a piece of paper" !ReplyDelete
ps. posting as anonymous as it doesn't like my wordpress url