Slashdot Mirror
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."
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.
www.tribalnetworks.org - helping tribal people around the world to own their own means of high-tech communications
Not really. NaN is: Not a Number.
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
Seriously, in elementary school a teacher of mine tried to tell us that 1/0 = infinity
Read up on the definition of division. 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).
Because mathematics doesn't deal with the real world. Physics does.
People take mathematical tools and models and apply them to the real world because they are useful. However, that usefulness is a lucky accident.
"Software is too expensive to build cheaply"
That behavior is a good thing. NULL is not 0 or an empty string -- it means "undefined". If you want 0, write 0. If you want "", write "".
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.
My other car is first.
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.
Parent is correct. It is truly mind boggling how terrible his reasoning is. You simply don't define infinity and -infinity as numbers. That is not what they are. Add this guy to the list of cranks http://www.amazon.com/Mathematical-Cranks-Spectrum -Underwood-Dudley/dp/0883855070
I say this report is Bullshit. What professor, after making a huge discovery, proceeds to teach it to children before presenting it at a seminar of his peers? If these children are his peers, then I suggest he merely drew a symbol and named it 0/0.
A Good Troll is better than a Bad Human.
0/0 should be a special case where dividing by zero actually yields a valid real number, and all other divisions by 0 are undefined.
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.
Wow. Looking over the guy's axioms, as soon as you introduce "nullity" the result of all of your computations is nullity:
- 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.
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.
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.
GAAH! MY PRINTER IS ON FIRE!!! PUT IT OUT! PUT IT OUT!
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 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.
Note to ACs: I usually delete AC replies without reading them. If you want to talk to me, log in.
Your "argument from intuition" is not a good one. Mathematics often comes up with nonintuitive results. In fact, that's the point, in a way. Mathematics is a set of rules and a language meant for re-expressing known truths in forms that lead us to realize new truths, all by rearranging things by the rules of logic. Here's an example: exponents were created to describe the number of times you multiply a number together with itself to get an example. 2^6 is 2 multiplied by itself 6 times. So what sense does it make to raise a number to a negative power? Well, ok, it's dividing one by that number a certain number of times. Or how about a fractional power, even more bizarrely? It turns out that devising rules that "make sense" often only make sense in the context of exactly the kind of discussion this guy has, purely within the realm of mathematics. I'm not saying his idea is good, rather that your argument is bad. :-)
Currently hooked on AMP