Professor Comes Up With a Way to Divide by Zero 1090
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."
NaN (Score:2, Insightful)
Dividing by zero is not a "problem"...... (Score:2, Insightful)
Oh, God... (Score:1, Insightful)
(These old folks know how to scream!)
testing, exception handling etc. (Score:5, Insightful)
This is computer programming ABC: you DONT allow undefined behavious to occur in your program! (especially if your doing MIL-STD Ada for avionics etc.) This guys 'method' is just a form of exception handling that any programmer with half-a-brain could implement.
Re:And this is important, why? (Score:3, Insightful)
Re:Umm... NaN? (Score:5, Insightful)
He proposes to define a new number that doesn't exist (or fit for that matter) in the current system.
But still it's useless, or at least I think it is.
100/0 != 10/0 != 1/0 != 0/0
but he uses the same identifier for all of them, so that would mean:
(100/0) / (1/0) = 1
That goes against the principle of:
infinity / (infinity - 1) != 1
Even I knew this was wrong as a 10 year old (Score:4, Insightful)
Read up on the definition of division [wikipedia.org]. If for a moment we ignore the "and the divisor is not 0" part of the definition, one of the basic principles of division is:
if a * b = c
then a / c = b, and b / c = a
A fundamental part of his explanation pivots on the following being true:
1/0 = infinity
-1/0 = -infinity
So, according to that, the following would hold:
if 1/0 = infinity
then infinity * 0 = 1
which does not work, for obvious reasons. This I told my teacher in 6th grade.
The real idea is that, for an equation 1/x = y, y approaches infinity as x approaches 0. At x=0, y is undefined, and that's all there is to it.
Secondly, the story promises one thing, and "delivers" another. It promises to tell you how to divide by 0, and instead tells you how to get 0^0 (which is based on the previously mentioned false premises). And the answer he gives on how to divide by 0 is that the answer is infinity, which it isn't! I'd fire the professor that has the gall of teaching this to kids (after probably being laughed out by his colleagues).
Re:Imaginary Numbers (Score:5, Insightful)
People take mathematical tools and models and apply them to the real world because they are useful. However, that usefulness is a lucky accident.
Re:And this is important, why? (Score:4, Insightful)
If you add a regular number and an undefined number, the result can't be defined. That's why 1 + NULL causes the entire operation to reduce to NULL. Makes perfect sense and is an important part of relational design.
Re:Sad, really... (Score:3, Insightful)
0*infinity=nullity
0 * infinitity becomes:
(0)*(1/0) becomes:
(0*1)/0 becomes:
0/0 = nullity
What about l'Hopital? (Score:2, Insightful)
I.e., it may well be that 0/0=a where a has a definite value? After all, any derivative is dy/dx=0/0.
That means to me that 0/0 is *really* undefined - may be this or that, depending on the circumstances; more information is needed, and assigning a specific symbol to it doesn't make much sense in the general case.
http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rul
Imaginary Numbers?! (Score:5, Insightful)
ALL Mathematics is COMPLETELY synthetic. That's the whole point -- that's the power of mathematics. You can define any set of rules, any set of axioms, any set of symbols, and start deducing. If the tools you need don't exist, you make them up. Nothing is more valuable in mathematics than a nice, clean, clear definition that increases the expressivity of math. Since math has no independent existence anyway, you can get away with pretty much anything so long as your new system has useful properties. Mathematicians with the guts to make things up as they go along end up with their names in textbooks and attached to great theorems, assuming what they made is conceptually useful (whether nullity is conceptually useful remains to be seen; a written description of the definitions would be nice).
Mathematicians that only do calculations that we already know about and are comfortable with? They're called accountants, and they have no friends. Seriously though -- since when did making up new ideas become a bad thing? I was under the (apparently mistaken) view that creativity was a praiseworthy trait.
Re:Sad, really... (Score:4, Insightful)
Re:Well, thats just nullty. (Score:5, Insightful)
crank shit (Score:3, Insightful)
Re:Well, thats just nullty. (Score:5, Insightful)
Wrong.
0/x gives 0. Always. And x/x gives 1. Always. Now, try for x=0... That gives 0/0 = 0 and 1 at the same time. That's why it's undefined, usually called NaN (Not a Number).
Anything else divided by zero can be defined as giving infinity or -infinity, which can be used in further calculations just fine, even coming to the correct result.
Example: The angle of the vector (1,0): arctan(1/0)*180/pi = 90 degrees. Works just fine. Not so for NaN, any calculation involving NaN will continue giving NaN.
Re:And this is important, why? (Score:1, Insightful)
Re:Sad, really... (Score:1, Insightful)
You said that 2 = 2*0*infinity = (2*0)*infinity = 0*infinity = 1
You could also say that 3 = 3*0*infinity = (3*0)*infinity = 0*infinity = 1 and so on
If you apply this formula enough times you'll notice that anything = anything*0*infinity = (anything*0)*infinity = 0*infinity = anything
This man is clearly a genius... he discovered that anything = anything!! Imagine the philosophical implications! For example, the universe can be whatever you want it to be (shit in a bottle of vodka) and visa versa!
Re:Dr. James Anderson's actual papers (Score:4, Insightful)
Unfortunately, that explanation seems to have been replaced by gibberish in the copy I just downloaded. Check it out:
So basically, the two NaNs have subtle semantics (much like his nullity) and don't have a catchy name or reuse a symbol that already means the golden ratio, therefore they're broken. In other words, they are defined, but he doesn't like the definition.Right. Now my airplane won't drop out of the sky, because the thrust calculation that used nullity as an input produced nullity as an output, in a way completely different from the one that produced NaN from NaN before. This new name and slightly different semantics magically mean the right amount of fuel will go into the engine.
He doesn't need to (Score:2, Insightful)
Only Chuck Norris can divide by zero.
Chuck Norris just looks at zero, and it divides itself.
He's just made "error" an object (Score:5, Insightful)
- the sum of anything and nullity is nullity (his axiom A4)
- the product of nullity and anything is nullity (his axiom A15)
- the reprical of nullity is nullity (his axiom A22)
So, his arithmetic is normal arithmetic, but as soon as you hit nullity anywhere, it's a black hole you can never get out of. All he's essentially done is take the "error state" and add it into the system as an object. You still can't compute anything you couldn't compute before. So yes, he has truly discovered NaN.
Re:Well, thats just nullty. (Score:3, Insightful)
He reasons thusly 1/0=+inf -1/0=-inf (already see the problem that +inf=-inf) and 0/0=nullity (hence nullity = +inf)
He then reasons that 0^0=0^(-1)*0^(1)=1/0*0 = 0/0 = nullity.
Now let's try that another way.
nullity*0=0/0*0=0/0=nullity.
But also if 0^0 is a number then 0^0*0^1 = 0^1 = 0 Thus nullity*0=0 hence nullity*0=nullity=0. This is a pretty clear contradiction.
In other words it's just dumb. Mathematicians are not idiots. We haven't missed something like this for thousands of years.
Re:Even I knew this was wrong as a 10 year old (Score:5, Insightful)
And for him it is true; he's defined infinity to have these values. He very specifically wants a fixed value for infinity.
Nor does this work. Division, in his system, is not the multiplicative inverse, but the reciprocal. So, for him: 1/0 = infinity implies 0/1 = 1/infinity, which does in fact meet our expectations.
Basically, what he's done with his system is come up with a (completely consistent, as far as I can tell from scanning from his website) framework where singularities now have a defined value, which means that all functions are defined everywhere on the real line (or the transreal line, which is what he calls his infinity-and-nullity supplemented system). Which is great, as far as it goes. But there's a big trade-off for this: there is now no longer a guarantee that if both f(x) and the limit at x of f both exist, that they will have the same value. The example he himself gives is the hypebolic tangent at infinity; the limit is 1, but by direct evaluation, it ends up being nullity. To get around this, he proposes a hierarchy of value determinations; a function is defined at a point by its transreal arithmetic value only if a different value isn't suggested by analysis. So tanh(infinity) would be treated as 1, even though working through the definition of tanh requires the value to be nullity in his system.
So in summary, he's defined terms so that division by zero is consistent and workable, but the price is that even relatively simple calculus becomes a lot more complicated. Nor is it all clear that transreal arithmetic will hold up with higher mathematics at all (when infinity is valued rather than defined by limits, how does cardinality work?). So I think he's got to a better job selling it than "it's better than NaN or having values undefined," because I can't see how it is.
Re:Argh!!! (Score:2, Insightful)
Basic math (Score:0, Insightful)
What's so amusing about that? Nah, what's amusing is the fact that exactly zero (0) slashdot nerds know that this is in fact the correct answer, and that the professor is wrong.
The answer to a / 0 is defined as the limit for a / x when x approaches 0. Some examples:
23 / 0 = lim x->0 (23 / x) = inf
-5 / 0 = lim x->0 (-5 / x) = -inf
0 / 0 = lim x->0 (0 / x) = 0
Simple as that.
Disclaimer: I haven't read the fine article, perhaps the professor meant something else.
Practical application of this nullity? (Score:2, Insightful)
I still fail to see how this helps people with pacemakers and computer related problems. Firstly any decent computer programmer making high integrity systems must care for situations where the divisor could be zero. Secondly there is no magical solution just by inventing a new concept. If your computer program should - even after your persistent effort - in an unforseen circumstance throw an divide-by-zero exception then just handle the exception and carry on.
Re:Well, thats just nullty. (Score:4, Insightful)
none. one nothing. Ten nothings. Twenty nothings. A billion nothings. Nothing * Anything = nothing.
Its not a number. Its a nonsense.
The first thing that crossed my mind... (Score:2, Insightful)
It's Not Rubbish (Score:5, Insightful)
That's why he's defined a new arithmetic - he calls it transreal - where division by zero is defined. The PDFs on his website clearly explain what he's done.
It isn't rubbish. In second year high school mathematics they had us "invent" our own arithmetic. We could define whatever operations we like (eg, a funny symbol that would multiple the left hand value by 2 and add it to the inverse of the right hand value) and then we had to prove whether the operation was commutative, distributive, etc. This guy has done the same thing but with a new "number" he calls nullity. He has defined what happens when you add a real to nullity, when you multiply a real by nullity, when you divide nullity by nullity, etc. It's an internally consistent number system.
It's interesting for grade schoolers because it gets them thinking about number theory. Instead of thinking "you can't divide by zero" they instead think "oh, well that's just a law for the real numbers, but I'm not constrained by real numbers, I can invent a number system where division by zero is allowed". That is far more insightful and creative than "you can't divide by zero". A child who grasps that concept has the potential to become a great mathematician. A child who merely parrots "you can't divide by zero" will become a bus driver or a computer programmer :-P
It's hard to explain abstract concepts such as number theory. Congratulations to him for making it look like fun.
Re:He's just made "error" an object (Score:4, Insightful)
Now that my original comment has been modded up, I should say, before anyone jumps on me, that this is not exactly NaN in the IEEE sense. In fact, this whole exercise seems to have been inspired by his own frustrations with the IEEE NaN. Better to say nullity is like "undefined", or some such thing.
Re:Imaginary Numbers (Score:4, Insightful)
Re:Well, thats just nullty. (Score:3, Insightful)
Re:Argh!!! (Score:4, Insightful)
Honestly, it's like Cantor never existed.
Re:Well, thats just nullty. (Score:3, Insightful)
Re:Umm... NaN? (Score:3, Insightful)
So why can't we do this for other items that are not a number, i.e. x/0?
Re:Basic math (Score:5, Insightful)
It's a bad example, because even outside of R, the left and right limits are not the same (one diverges to minus infinity and the other plus infinity).
lim x->0 (23 / |x|)
is better. It is undefined because it exceeds R, one could technically define a set of numbers which includes +=infinity, in which division by zero would be defined.
Re:Well, thats just nullty. (Score:5, Insightful)
Yes, actually it is, and there are different sets of rules (aka axioms) that are used. For example, Euclid chose to include the Parallel Postulate [wikipedia.org] among the axioms that define his geometry, but there are various well-developed -- and useful! -- non-Euclidean geometries that assume the parallel postulate is not true. There are many branches of mathematics that modify what most would consider the "normal" rules in various ways. Many of them prove to be useful in the real world, too.
Mathematicians realized a century ago that their work is a discipline of arbitrary rules, and that none of their theorems have any inherent real-world truth or falsehood. Math is simply an abstract model. By choosing the right set of axioms one can create a model that maps well onto various aspects of reality, making it useful for physics, engineering and much, much more. Sometimes the common rule set doesn't map well, and even physicists and engineers use the alternative rule sets mathematicians have devised.
This concept of "nullity" isn't something that mathematicians would call wrong. For it to be wrong, it would have to be inconsistent with the results of whatever other axioms Anderson has chosen to use. What mathematicians would call it, however, is an old, uninteresting idea. There have been many others that postulated a placeholder "value" for infinity and explored the results of that assumption. Some of the results are even occasionally useful in simplifying useful calculations. And sometimes the alternative system produces results that don't map well onto reality, and the distinction between the cases is well-explored and well-understood.
I may be stating that too strongly, though. It's possible that Anderson has adjusted his definition in a way that makes it useful for a broader set of problems. Honestly, though, I doubt it. This is thoroughly plowed-over terrain.
I think it's most likely that Anderson has discovered some specific, important problems in optics(which involves some very high-powered mathematics, BTW, much more so than most engineering disciplines) that can be simplified by postulating a nullity, and that he published the work in an appropriate journal to an appreciative audience.
Re:It's not? (Score:3, Insightful)
You persist in thinking that there is "right" math and "wrong" math. That is not true. Math is just a big pile of abstract formalisms. As long as they're self-consistent they're correct, by definition. What most people learn as math is just a small subset of mathematics which is relatively simple and maps well onto common, everyday reality.
If your goal is to teach kids how to count widgets and manipulate dollars, then there's no sense in exploring broader mathematical ideas. If your goal is to create budding young mathematicians, then it's a really good idea to expose them to the idea that math is both bigger and more malleable than the set of ideas their teachers are going to present.
The best, of course, is to do both. Give all of the kids a good grounding in ordinary arithmetic, geometry, algebra and (ideally) calculus since everyone needs arithmetic skills and many, many people need algebra and calc (including many who get by without it). While you're at it, though, throw in an occasional bit about the broader sweep of what mathematics is and what mathematicians do. Try to avoid confusing those that don't have the ability to think about such abstractions, but by all means exercise the potential of those that do, because there are more of them than you might think.
Finally, your crack about mathematicians being isolated from reality just shows that you don't know any mathematicians.
Whoever modded this - get a grip! (Score:2, Insightful)
Re:Simultaneous Equations (Score:2, Insightful)
Anyway, ambiguity can be fun. Perl modules need to evaluate to true, so people usually end them with "1". I usually wrote "3!=6", which is doubly true
Re:Argh!!! (Score:3, Insightful)
Nullity already defined (Score:2, Insightful)
Re:Well, thats just nullty. (Score:2, Insightful)
The article seems lacking in detail. By contrast, proper mathmagicians can't even get to 2 without using a long and difficult proof.
My 2 cents: wouldn't 0/0 be the set of all complex numbers? If x = 0/0 then 0 = 0x would work for any x.
How can this be a ring? (Score:4, Insightful)
Unfortunately, he just complicates things, because he doesn't define how the + and * operators map up with it (nullity + a = ?)... if he doesn't then he breaks assoc/commu/trans properties (no longer a field then). And of course that number we need additive/mult inverses which may require nullity-prime, and so on, and he's just going in circles.
Re: Re:Argh!!! (Score:3, Insightful)
Re:NO!: do sneeze at it (Score:1, Insightful)
A theorem that guarantees the existence of a nice continuous version of a function does not help in any way when you're trying to do arithmetic with a non-continuous function. Do you know what arithmetic is? Do you know what a computer is? Do you realize that you're reading the work of a computer scientist discussing arithmetic for the specific purpose of proposing that computers use a different abstract machine model?
Re:Argh!!! (Score:5, Insightful)