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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."
I hate to put it this way, but "It'll make sense when you're older". And by older, I mean when you take a higher math course. What is the square root of -1 equal to then? Nothing? Something? Saying it's "imaginary" is merely a construct that allows us to muck with things. We could say they're "happy fun times" numbers, with the symbol "hft", and it'd mean the same thing.
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:
Submitter couldn't be bothered to do the research, but there is a paper written by this guy about the concept.
"Elmo knows where you live!" - The Simpsons
You simply don't define infinity and -infinity as numbers.
_ line
Well, not Reals, at any rate:
http://en.wikipedia.org/wiki/Extended_real_number
http://en.wikipedia.org/wiki/Real_projective_line
Tweet, tweet.
Actually, in his own paper here:
a chineVIII.pdf
http://www.bookofparagon.com/Mathematics/PerspexM
he agrees with closely what you just said. It says nullity*14 = nullity, he's created a new mathematic Field. It still might be a load of hot air but this paper is at least more rigorous than the video's 0/0 = nullity craziness.
If you don't know what a Field is its kind of like a different universe for numbers with different rules than the one's we're taught in school. Another example of a Field would be discrete mathmatics which is used for encryption.
The answer to a / 0 is defined as the limit for a / x when x approaches 0.
So you've proved that f(x) = 0/x is continuous?
lim x->0 (23 / x)
lim x->0 (-5 / x)
Neither of these exist.
Play Command HQ online
To be fair it's not entirely uncommon for mathematicians to invent new concepts. Take as the primary example the square root of -1, this is the imaginary number i. So having a symbol to designate dividing by zero quite sensible, does it help the maths, well no because once you divide by 0 algebra stops making sense eg.
1 x 0 = 0
(1 x 0)/0 = 0/0
and
2 x 0 = 0
(2 x 0)/0 = 0/0
It then follows that
(1 x 0)/0 = 0/0 = (2 x 0)/0
so you have
1 x 0/0 = 2 x 0/0
cancelling the x 0/0 you have
1 = 2
(there are more elegant proofs than this i just can't remember them this morning)
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
If you speak about limeses, then it depends how you go toward some value (toward 0 in this case).
For instance, both functions f1(x)=sin(x) and f2(x)=x are 0 for x = 0, but
lim x->0 (sin(x)/x) = 1, as we know.
If you take function like f1(x) = x*sin(x) and other one f2(x) = x then
lim x-> f1(x)/f2(x) = 0.
In these two cases, "0/0" have different values.
When you use division in limeses, the path you take is important, i.e. functions that describe in which way you go toward 0. That's why other posters mentioned continuity and other stuff related to functions, and not related to numbers.
Big breakthrough would be to solve lim x->0 f1(x)/f2(x) for f1(x) = 0, f2(x) = 0.
No sig today.
The problem isn't that people haven't figured out ways of dividing by zero, the problem is that there are many different ways in which you could reasonably define division by zero, and they are not mutually consistent. Wikipedia lists some of them.
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.
"one could technically define a set of numbers which includes +=infinity"
Technically you could not do this. Remember, infinity is not a number, it is a concept meaning an unbounded limit. There are rules for including it in algebraic equations, but it is still not a "number."
Anyway the proof as I know it is this: Define 0 as a number. Define a successor function which takes a number as input and produces a number as output. Then start defining some labels like 1 (doesn't really have to be 1, could be the Symbol formerly known as Prince... just a label... still the same crazy music genius... this, it would be nice if were explained more...) is the Successor of 0, 2 is the Successor of the Successor of 0, 3 and then 4 in the same way. Then finally define + as the following construction: 0 + any number = that any number and S(x) + S(y) = x + S(S(y).
2 + 2 = 4
S(S(0)) + S(S(0)) = S(S(S(S(0)))) by definitions above.
S(0) + S(S(S(0))) = S(S(S(S(0)))) by the second rule of +
0 + S(S(S(S(0)))) = S(S(S(S(0)))) again by the second rule of +
S(S(S(S(0)))) = S(S(S(S(0)))) by the first rule of +
QED
Anyway, ask some 6 year old who knows how to count on their fingers... they'll show you that (holding two sets of fingers on either hand and then counting the "successors" by dropping fingers as they go.)
People who quote themselves bug the crap out of me -- Me.