I expect that will be fixed shortly. The middle line is written X years ago and set X years ago and thus contemporary. I thought it was a bug at first. It says "Stories written X years ago and set 2X years ago.
In reply to John Gabriel: How is that not well defined? They are indispensable in quantum mechanics. You simply cannot do quantum mechanics without them. The wave functions are in general complex. Not even sines and cosines. The Schrodinger equation, the basic equation of QM, has an i in it.
And the momentum operator has an i in it. There is no way around this. And quantum mechanics may well be the most successful theory of all time. Certain quantities the gyromagnetic ratio, e. Complex numbers are also immensely useful for electrical engineers and electromagnetism in general.
Complex theory is required to prove the fundamental theorem of algebra. And it can be used to evaluate certain real definite integrals.
Square roots apply to all complex numbers, real or imaginary. But with the series you can easily generalize it to complex numbers. The problem is that you refuse to allow for legitimate generalization. Using your general argument we would be stuck with the natural numbers or perhaps whole numbers.
If you divide an object by 2, you now have two objects! But they are a very useful generalization. Money a balance comes to mind, in addition to science. Rational numbers allow division except by 0. They are quite legitimate. They allow for a measure of the wave function in quantum mechanics.
The denominator is a real number, and division by a real number is quite well defined. The reason complex numbers are not ordered is because they fail to satisfy the criteria for an ordered field. It has nothing to do with division. One of the properties of an ordered field is that any number squared is a positive number.
Therefore, i is not a positive number, and therefore the field of complex numbers is not ordered. Therefore, i is not a negative number.
Imaginary numbers are neither positive nor negative. Now for the 1 thru 4 bit. So sqrt -1 is multivalued and can be i or -i. With positive real numbers there is a natural choice for square roots — the positive root.
More generally you use the log-based definition except for a base of zero. Sticking with positive real numbers for the base, and real numbers for the exponent, you get single-valued functions and the exponent rules all work fine.
I see no problem with Eq. And since i is neither positive nor negative, this rule cannot be used anyway. The exponent rules work for complex numbers if you work in polar form except for a base of zero, where the power must be a nonnegative real number.
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