What are imaginary numbers?
An imaginary number is one whose square is a negative number. Of course, no "real" number that you know, when multiplied by itself, will give a negative number, since two negative numbers multiply to make a positive one.
For example (and "x" means "times" in the following), 1 x 1 = 1 and (-1) x (-1) = 1 so both 1 and -1 are "square roots" of 1 - that is, they both give 1 when multiplied by themselves. If you ask what you'd have to multiply by itself to get -1, you find you're out of luck, so what mathematicians and physicists do is to invent a symbol "i" and just declare that i x i = -1. Here "i" isn't any number at all that you'd normally think of as a number, so it's said to be "imaginary." All imaginary numbers are then any real number times i. For example, 3i is imaginary, and it squares to -9 since 3i x 3i = 3 x i x 3 x i = 9 x (i x i) and 9 x -1 =-9.
If this looks like it was completely made up, then you have understood the point.
It turns out, rather amazingly, that a lot of our world is describable in very simple terms with very simple equations if one uses this idea of imaginary numbers, and there's a nice geometric interpretation.
Imagine you're standing 10 feet east of a landmark. Multiplying that distance by 1 leaves it alone. Multiplying it by -1 means being 10 feet to the west of the landmark and multiplying by -1 twice brings you back to 10 feet east. Multiplying by i turns out to means putting you at 10 feet north of the landmark (i.e. a 90 degree turn to the left) so that two multiplications by i get you to what one multiplication by -1 did. In other words i x i = -1.
It takes a little practice, but it turns out that these imaginary numbers are just the thing to describe things that happen in a plane instead of along a line, so in a sense they're no more imaginary that turning 90 degrees!
Dr. Knowledge is written by physicists Stephen Reucroft and John Swain, both of Northeastern University. E-mail questions to drknowledge@globe.com or write Dr. Knowledge, c/o The Boston Globe, PO Box 55819, Boston, MA 02205-5819.
