Ask a simple question
We wind up unexpectedly far from home.
At a fairly early stage in our mathematical education we were introduced to the idea of raising to a power: x2 means multiplying x times x, x3 is using x as a factor three times and x4 four times. At the time, some of our consultants were inspired to look at the series 11, 22, 33, 44 and found that it got very big very quickly. A bit later in algebra class we learned how to work with fractional exponents, which would allow us to fill in a smooth curve for xx from almost zero out to as big as our piece of paper would admit. In calculus, sometimes students are asked to find the limit of this curve as x gets close to zero (00 itself has no meaning). But that’s as far as anyone really needs to go with this particular bit of math; the function doesn’t seem to appear in any problem in physics, and has attracted no apparent attention from mathematicians.
But then one of our consultants asked the question: what does this curve look like when x is negative? Well, figuring out -1 is easy, and -2, but as soon as we ask about the points in between we’re forced into complex numbers, those that include imaginary numbers. There’s no way around them. Imaginary numbers lie in a strange sort of space at right angles to the ones we can use to count sheep and tax deductions, and most people are uneasy there; but if you ignore it, your curve is full of holes and inconsistencies. And then we realized that even the positive side of our curve should have included complex numbers. Simply asking the question led to a more complicated area of mathematics (which, to be fair, has been very fruitful).
But, to be complete, we have to consider what happens when x itself is complex. Then things get much more difficult to see, if not calculate. A complex number is a two-dimensional objext; a function like xx is thus a two-dimensional surface in a four-dimensional space, and none of us can picture what that looks like. (An early attempt to draw just three dimensions of the problem turned out to be very misleading.) Well, the consultant who asked the question studied a complex-function book, and found that one way to approach the problem was to take a figure in the input complex plane (like a circle or line) and see what it came out to in the output plane. The functions in the book made interesting and insightful graphs.
He tried it. A simple circle came out to be a double-looped curve; lines turned into spirals, growing or shrinking. A single line broke into an infinity of different lines, and those through the origin in the input spiraled around unity in the output, more and more rapidly as they approached it. A single input point produced an infinity of output points, and vice versa. We’re still not sure what it looks like, but it threatens to make us sea-sick.
The moral of the story is that asking a simple question in mathematics is dangerous. It can lead you a very long way from home.
