Ramble, Part 39: The Complexity of Numbers

[stoutfellow]

Let me recap some things I discussed in the very first Ramble posts. The concept of "number" has two distinct and not entirely compatible origins: counting, which gives rise to the positive whole numbers, and measuring, from which are derived the positive real numbers. The two constructions have been melded together since time immemorial, though, with consequences which I've talked about at some length.

As time passed, various lines of mathematical study made it clear that the notion needed to be extended. First in India, then in the Muslim world and finally in Europe¹, the notion of negative numbers arose; somewhere along the line, complex numbers put in an appearance. These extensions, though, were haphazard, tentative, and mistrusted. Around the turn of the nineteenth century, though, European mathematicians finally confronted these ideas, in a variety of ways, and the next two or three Rambles will discuss this.

Even though, as mentioned before, Rafael Bombelli had shown that complex numbers were absolutely essential to Cardano's solution of the general cubic equation, they were not generally accepted for long afterward. (Descartes was uneasy even about negative numbers; where we would say that an equation has -4 as a root, he would say that 4 was a "false root". This did not prevent him from working with negative numbers, but it does suggest some discomfort with them. How much less accepting would less adventurous souls than his be?)

The value of complex numbers, though, was quite apparent; by Newton's time, the Fundamental Theorem of Algebra - that every nonconstant polynomial has a complex root - had been conjectured, and several mathematicians, including Newton himself and (of course) Euler, had wrestled with proving it. (Euler approached complex numbers with the same panache he applied to everything in mathematics. It was he who gave the square root of -1 its customary name, i, and he went as far as to insert them into his beloved infinite series, thereby deriving his famous equation e^ix=cos x+i sin x.)

Part of the problem with the acceptance of complex numbers had to do with the lack of a convenient visualization: it was difficult to develop any intuition as to their behavior (and the value of such intuition should not be underestimated). Oddly, it was a pair of non-mathematicians, working independently, who found the way out of this impasse. Caspar Wessel, a Norwegian surveyor, and Jean Robert Argand, a Swiss bookkeeper, both suggested representing the complex number a+bi as the point (a,b) in the coordinate plane. Wessel published first, but his paper was largely ignored; Argand had somewhat more success, and the victory of the idea was assured when the great Carl Gauss came out in its favor and, perhaps characteristically, claimed the idea for his own - not entirely without justification, but, well, that was Gauss for you. (More on that later.) Many sources refer to the complex numbers, so conceived, as the Gaussian plane; the sophisticated refer to the Argand plane. Poor Wessel, who really was first, has been largely forgotten.

The idea paid immediate dividends. Using it, Gauss was able to develop the first satisfactory proof of the Fundamental Theorem. The proof was geometrical in nature, which Gauss found displeasing; he derived several more proofs over the course of his career, but was unable to rid them of the geometric taint. (It is now generally accepted that geometric ideas must enter in, as the complex numbers, like the reals, are in some sense essentially geometric rather than purely algebraic constructions.)

Gauss' proof works as follows. Let f(z) be a complex polynomial. Write z=x+i y, and break f(z) into its real and imaginary parts: f(x+iy)=g(x,y)+ih(x,y), where g and h are now real polynomials of two variables. Then f(z)=0 if and only if g(x,y)=h(x,y)=0. Now the equations g(x,y)=0 and h(x,y)=0 define a pair of curves in the plane, and Gauss was able to show that these curves must intersect. The point of intersection is a complex zero of f.

The geometric model of the complex numbers was to inspire further developments of interest, but the scene has to shift to the British Isles first, where the tide was running in a decidedly different direction. That will be the subject of the next Ramble.

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1. I omit China from consideration; I don't know enough about Chinese mathematics, and in any case it followed a largely separate line of development from the rest of Eurasia.

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Comments

[countrycousin.livejournal.com]

A Plane and Simple Idea

boo hiss ;<)

I only recall "complex plane" for the geometrical representation. That is what was used in Ahlfors, but the instructor very well may have more edifying - that was a long time ago. If I was ever exposed to this background, it had long gone - thanks for doing this!

[stoutfellow.livejournal.com]

The phrase "complex plane" is itself slightly misleading, as - from the point of view of the complex numbers - it is actually one-dimensional. In certain branches of geometry, it's referred to as the complex line....

[countrycousin.livejournal.com]

*tries to wrap head around concept of complex number as a line*

*goes off to find Excedrin*

I fear I am culturally deprived.

[stoutfellow.livejournal.com]

Let me try to explain - or at least point in the direction of an explanation.

What's going on is metaphorical. We speak of the real line, because we can identify each point of the Euclidean line with one and only one real number. In the same way, we can identify each point of the Euclidean plane with one and only one real number: the Euclidean plane is to the complex numbers, as the Euclidean line is to the reals. Metaphor, then: if we say "the real line", can we not likewise say "the complex line", meaning the Euclidean plane (conceived in a particular way)?

It's a flawed metaphor, of course; the Euclidean line has many properties that the Euclidean plane does not. The only question is, is the metaphor useful? That depends on what you're trying to do; and there are branches of geometry in which the metaphor is sufficiently powerful to be worth accepting (though not, of course, without recognizing its potential to mislead or confuse).

[countrycousin.livejournal.com]

OK, thanks. The headache came from the properties that I associate with "line" that the complex numbers do not have. I can think of them as one "variable" and I'll take your word for it that the metaphor has its uses.

[stoutfellow.livejournal.com]

Aak.

we can identify each point of the Euclidean plane with one and only one real number

That should be "complex number".