That's really an interesting analogy, RWP. But I don't see the practical application to our present inquiry. (Maybe you could help me with that.)
I had thought that imaginary numbers are "abstractions" mainly used for practical reasons, as a (very ingenious) way to allow square roots of negative numbers to figure in various types of mathematical equations, which facilitates a wide range of scientific applications; e.g., plotting positions in modeled spaces, etc.
In other words, It seems imaginary numbers facilitate what we already know we want to get done -- they are means to an end. But what is the end, or goal, or purpose? The need to posit imaginary numbers seems to suggest that we already know what that end or goal or purpose is before we begin. So, by analogy, is there a "cosmic knower" who would use imaginary numbers in this manner, and thus produce a singularity? (Is Hawking the "cosmic knower" in this sense?)
Do imaginary numbers really figure in nature -- that is, are they discoveries by man of what is already "there"? Or are they inventions or artifacts of man, useful tools or "machines?"
More questions than answers, as usual, RWP! Thanks so much for writing.
Real numbers form a group under most arithmetical operations - that is, multiply, add, divide, square, subtract real numbers, and you still have a real number. Just don't take a square root. :-) When you say a physical quantity can be described as a real number, what you're postulating is what group theorists call an isomorphism; the set of real positive numbers can be mapped one on one with the possible masses of an object, for example. So mass is a real, positive quantity.
Complex numbers have more interesting and inclusive properties than the group of real numbers. You can take square roots of complex numbers and the answer will always be a complex number, for example. And many physically meaningful quantities are isomorphous with the complex numbers - refractive indexes, alternating currents, etc.. There is an even more general group, discovered by Hamilton, called the quaternions. And all of these, in modern math, comes under the category of group theory, which tells you what possible kinds of number sets and algebras can self-consistently exist.
So, in standard physics, time is a real number. But any real number is by definition a complex number with a zero imaginary component; the set of real numbers is a subset of the complex numbers, which is a subset of the quaternions, which... It's an interesting question to ask (in my very humble opinion, in this case) what would happen to physics if the imaginary component of time were non zero, that is, if time were more complicated than we think it is. It's an interesting question I can't answer, but maybe Hawking can.
Not different from any other mathematical entity. Perhaps calling them 'imaginary' put an unfortunate color to them. They are real and valid, not a metaphysical hypostatization, but a logical rather than psychological extension to our primitive 'count' numbers.
No more so than integers or differential equations.
Complex (real and imaginary joined) numbers have several useful properties. For an example, if one solves x^2=2 (going back to Pythagoras), one finds that there is no rational p/q equal to x, so x is of a different kind than fractions. However, x^2=3 needs a new number, just attaching x from x^2=2 doesn't help solve x^2=3. Attaching all the (infinitely many) x^2=p for all prime integers p, doesn't solve the system x^3=2. It gets worse, attaching all the roots of all the integers gives neither pi, nor e. Eudoxus (and Dedekind and Cantor) gave complete discriptions of the real numbers, but these descriptions are rather long and technical.
With imaginary numbers, things are a bit different. One introduces the solution to x^2+1=0, (named i for this note.) Then the solutions to x^3+1=0, x^245+24x^44=0, and all polynomials in x have a solution involving i. Even equations with complex coefficients such as (3x+2i)x^5+7=0 have solutions in complex numbers. This is the Fundamental Theorem of Algebra.
Complex numbers are not just pairs of reals. They are pairs of reals with special rules for arithmetic. They form a field. The rules are (a+bi)+(c+di)=(a+b+(c+d)i) and (a+bi)*(c+di)=(ab-cd+(ad+bc)i). One gets the same result by just using i^2=1 and treating i as a "number."
These complex numbers describe electrical and magnetic fields and are useful in Fourier series because of the identity e^(ix)=Cos(x)+i*Sin(x) (Euler's relation.) Quantum mechanics requires complex numbers for its descriptions. For example, probabilities occur in QM as squares of numbers (actually squares of averages over wave functions) but the numbers are complex (the "square" of a complex number (a+bi) is actually taken as (a+bi)*(a-bi)= a^2+b^2, a-bi is called the conjugate of a+bi.) When two "waves" add in quantum mechanics, the complex numbers describing are added and the square of the result taken to give probabilities. (Classically, one would just add the probabilities.)
Complex numbers are essential in analyis. For example, take a simple function f(x)=1/(1+x^2) with real x. A power series for this function will only converge if |x| < 1. This is because there is a zero at x=i (or x=-1.) Things happening in the complex numbers are intruding onto the real line.
Imaginary numbers are just as "real" (in the ontolgical sense) as real numbers (or just as imaginary.) The terms are unfortunate as ordinary language uses the words "real" and "imaginary" differently. (Not surprisingly terms like: group, field, radical, or, implies, if, then, else, not, and, root, function, etc. have technical meanings that don't correlate well with ordinary usage.)