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Professor Comes Up With a Way to Divide by Zero
54mc writes "The BBC reports that Dr. James Anderson, of the University of Reading, has finally conquered the problem of dividing by zero. His new number, which he calls "nullity" solves the 1200 year old problem that niether Newton nor Pythagoras could solve, the problem of zero to the zero power. Story features video (Real Player only) of Dr. Anderson explaining the "simple" concept."
Uh... are you joking?
Imaginary numbers (specifically, complex numbers, which consist of a sum of a real and an imaginary number, and which comprise the "complex plane") are INCREDIBLY important in the "real world."
I'm just a chemist, not a mathematician, but I am well aware that imaginary numbers are critical in the Fourier transforms used every time I take an IR or NMR spectrum.
Ever do electrical engineering? Circuit analysis is made a great deal easier when you can treat circuit elements in terms of complex numbers. All that "impedance" stuff you hear about capacitors and the like that makes it possible to apply Ohm's Law to LRC circuits.
These also are not merely made up properties, they are fundamental to mathematics and thus (if one believes that math is the language of the universe) physics. For example, certain integrals necessarily yield imaginary results. These integrals are not of some ethereal interest, but appear throughout quantum mechanics. This is why the amplitude of a wavefunction (used, for example, in molecular modeling that allows for practical achievements like better medicines) is not the square of the wave function (or, for that matter, its absolute value) but the product of the wavefunction and ITS COMPLEX CONJUGATE.
If you'd like more examples of the utility of complex numbers and other "random rules," check out Boas' "Mathematical Methods In The Physical Sciences."
At first, numbers were integers - what you could count on your fingers. (N) Later on, numbers were fractional - in order to express the sharing of things. (Q) Later on, numbers were negative - in order to express debt. (Z) Even later on, some numbers were found not to be fractionar (the first proved was square root of 2). Enter R However, not every polinomial equation has its solutions as real numbers (see x^2+1=0). The solution to this equation was named i, with the property that i squared is -1. It was called imaginary because no real number had such property, and it is as real as a figment of your imagination ;)
While other real numbers can be aproximated by integers, negative integers and fractional numbers (with better and better accuracy), i has no aproximation in any of the previous pools of numbers.
In engineering, a useful aproximation for pi is 3. There is no aproximation of i as an integer.
Infinity isn't a real number. Ergo, it cannot be the limit of a sequence, as the definition of a limit include the priviso that it is a real number.
You can only perform the substitution lim x->a f(x) = f(a) when f is continuous at a. f(x) = 1/x is (very trivially) not continous at a = 0.
Damnit, why is this sort of thing spilling over from sci.math now?
After all, I am strangely colored.
The first paper he describes as:
The second paper he says:
The limit of a constant over x as x approaches zero would depend on which direction you're approaching x from. For 23/x, if you approach 0 from the left, you get -inf, and if you approach it from the right you get a positive inf. Really, though, the behavior is better defined as an unbounded number approaching positive or negative infinity.
lim x->0+ (1/x) = inf
lim x->0- (1/x) = -inf
I think the GP was refering to the hardware level, not an abstract software layer. Where traditonal computers, even those with modern math extensions dont know what an imaginary or complex number is. Normally, two floating point values are used to represent complex arithmetic, however its not a native operation, and still requires some software logic to be accomplished.
if a=b, then (a-b) = 0. going from the fifth line to the sixth line, when you divided out (a-b) from both sides, you were, in fact, introducing a nullity.
do not read this line twice.