Posts Tagged 'Symmetry'

Imaginary Numbers are Real

Mathematical wisdom from xkcd. Warning: contains swears.

Imaginary and complex numbers are handicapped by the name we gave them. “Imaginary” has obvious and bad connotations: it implies an object made up, perhaps not useful; “complex” similarly seems to argue the numbers are too hard to use. As with so much in math and physics, however, the names are historical, and complexity is context-dependent. In particular, imaginary numbers have an obvious reason for existence—the square root of a negative real number—and as I noted in an earlier post, complex numbers are incredibly useful. While it’s true that the algebra of complex numbers is a little trickier than “ordinary” algebra, it’s not phenomenally harder. The order of multiplication doesn’t matter, as it does with quaternions, Clifford numbers, or Grassmann numbers. In fact, for some applications like electrical engineering or the processing of signals, complex numbers turn out to be easier to use than real numbers! After all, if a particular type of math makes something simpler, it might still just be a tool, a convenience. While it’s true that sometimes complex numbers are mainly a convenience, because of their deep connection to geometry, they are as real as any other mathematical device.

Rotating a square using complex numbers. For details, see my earlier post, “The Most Striking Equation in Mathematics“.

The earliest history of complex numbers ties them to solving algebraic equations, like quadratic equations: (insert example here) However, mathematicians realized soon that they could be used to represent points on a plane. In fact, today is the birthday of mathematician Caspar Wessel who characterized the complex plane. Thanks to this discovery, we know that anything that can be represented by a two-dimensional mathematical object (coordinates) can also be written as a complex number. We can rotate objects in two dimensions (see the figure at left) and describe waves. While complex numbers aren’t strictly required for either of those operations, they make our mathematical lives easier by their existence.

I used to give a talk in graduate school called “Imaginary Numbers are Not Real”, a title and concept I borrowed from Stephen Gull, Anthony Lasenby, and Chris Doran. The talk and paper addressed Clifford algebras, arguing that every imaginary number in our math corresponds to a geometrical concept. I still agree with most of what I said back then, but I’ve changed my rhetoric a bit: I now say imaginary numbers are real, in the sense that they provide meaning to the results of real physics experiments. If the quantities they characterize exist in nature, I say complex numbers are as real as any other mathematical entity—not as a Platonic ideal, but as a true description of the Universe in which we live. Continue reading ‘Imaginary Numbers are Real’

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