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."
So much for my $200 calculator.
--- Grow a pair, liberals... stop letting the Republicans bully you!
His new number, which he calls "nullity"
:-)
Well, thats just nullty.
Seriously though, as I understand it, this is simply another mathematical structure that allows a different scalar much like a real projective line, right? If that is the case, then there is nothing really new here and there can be no application or definition with real numbers or integers. Alternatively by interpreting this as a commutative ring, one might be able to extend this to where division by zero does not always get you in trouble, but the precise interpretation of "division" is fundamentally altered. This too is not a new concept.
However, all of that said, I am a bioscientist and my math skills are not as strong as a formally trained mathematician, so I will defer to those here who are stronger mathematicians than I if this interpretation is incorrect.
Visit Jonesblog and say hello.
The professors at 'Rithmetic State were non-plussed upon hearing the news.
Is it just me or does it sound like he thinks he's invented the NaN?
There's zero comments yet. Wonder how many comments that is per poster
I can make up numbers too...
What he did was assign the previously "undefined" integer with a defined symbol that means the same thing. Infinity in both directions.
While interesting, the concept has little use.
From the article "Imagine you're landing on an aeroplane and the automatic pilot's working," he suggests. "If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead.".
Now, instead of getting an error message, the computer give a 0 with a line through it, and THEN an error message.
--sig fault--
mod original post up by 0/0 points :)
-- "Genius is 1% inspiration and 99% perspiration" - TAE --
Only Chuck Norris can divide by zero.
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
Yet Another NaN? ;)
--I thought I was wrong once, but I was mistaken.
It's sad that he teaches math and thinks this is a worthwhile concept.
For just one example of why it sucks, he BEGINS by defining: (infinity) = 1/0 and (-infinity) = -1/0.
My conclusion: (0)*(infinity)=1
So 2*0*infinity = 2*1
So 2 = 2*0*infinity = (2*0)*infinity = 0*infinity = 1
And once you know that 2 != 1 and 2 =1, it turns out you can prove quite a bit...
Total nonsense, and the BBC is encouraging it. *shakes head* Although, I've got to say, it's nice, for once in my life, to deservedly be a smug American.
Think! It ain't illegal yet!
George Clinton
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."
Helpful little hint from the end of the video:
Yeah. It was that simple.
I'm just reminded of that proof from way-back-when that 2 = 1:
All this guy has done is provide another little fun "proof" that you can use to win bar bets. "Betcha I can divide by zero..."
The heavens do not fall for such a trifle.
Mr. L'Hopital would have something to say against this.
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.
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"
Note: IAAM(athematician). You pose a good question. The game in mathematics, though, is not to "make up random rules so that something that occurs to them suddenly works". It's (broadly speaking) to make up new rules which are completely consistent with all the old rules which allow us to understand a previously mysterious example. This is where "imaginary" numbers succeed tremendously, and "nullity" fails miserably. See my post downthread for why nullity sucks.
"Imaginary" numbers are just the "thingys" which are solutions to polynomials. I.e., mathematicians find it useful to have an answer to the question "for what values of x does x^2 + 1 = 0?" The answers are useful, even though they aren't good at measuring length or breadth or depth or other one-dimensional concepts. They're useful because they allow mathematicians to develop a theory which has answered questions which couldn't be answered before. This is true even though both the question and the answer both lie in the realm of real numbers. Should there be an answer to every question of this type that doesn't use complex numbers? Perhaps, but it certainly doesn't have to be pretty, or easy to discover. Often the shortest path to a "real" truth lies on an "imaginary" line.
Think! It ain't illegal yet!
George Clinton
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:
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.
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
Seriously though this is the sort of thing that you don't want to sneeze at, it can sound both inane and brilliant. Anderson is not such a crackpot, I found a presentation of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge."). He seems to be a geometer stating his perspective in the first line of that presentation: "Aims: To unify projective geometry and the Turing machine".
He's a geek hero! Who knows if his nullity will end up just NaN with a British twang or the next best thing to sliced bread and i?
I was unable to hear the realaudio casts but from Book of Paragon, The Perspex Machine (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations (a mathematical treatise with applications to computer vision and robotics) just glancing I'd have to say the guy seems to be a real mathematician, geek and philosopher-king. I don't know if he's up there with Newton but he at least deserves an honorable mention for his wonderfully witty (and to me as yet inscrutable) naming of the Walnut Cake Theorem (see page 10 of Perspex.pdf). It seems that he was motivated to create nullity in order to make reliable advanced computers that would not barf when asked questions about the universe, and to him "Not-a-Number" is vomit. I'd say read some of his stuff before assigning him to the 9th Hell. Would like to hear what any mathematicians or other people with brain cells over the age of 12 have to think about it. It's okay if he reinvented something but it appears he is trying to make a machine that can handle infinities and other tough numerical concepts with ease, and that's worth something. Oh, that and his quantum computer looks neat.
0/0 is special, explained:
Think of a division as the reverse of multiplication:
6 / 2 = 3, which means 3 * 2 = 6
With a division by 0, this does not hold:
6 / 0 = x, there is no possible x for which x * 0 = 6
X can be no real number
However, 0/0 is different:
0 / 0 = x, but no matter what you fill in for x, x * 0 = 0
X can be any real or imaginary number, 0 * x is always 0
This is why A / 0 has no solution, unless A = 0, then A / 0 does have a solution, an infinite number of solutions in fact: all numbers are a correct solution.
This professor didn't invent it by the way. He just seems to be the first to bother explaining it to school children.
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.
This was a question posed in a book I read a while ago, by some reknown mathematician...for all his accomplishments, he couldn't help but wonder...was any of it really helping to describe the universe better and broadening our knowledge of it (thus, a discovery), or was more of it simply a figment of his stretched imagination?
So Nullity may now 'officially' mean n/0 but what does it mean really? Is it just another term for, say, infinity or undefined?
I will never forget when I was about 8 years old going up to the adding machine in my grandfather's home office. It was about twice the size of a toaster and made of that old typewriter metal. It looked like it weighed as much as a car and had probably cost as much new. Just to see what would happen I entered '0', '/' and '0'. Without hesitation it began producing line after line of '0', '0', '0' on the paper tape accompanied by a cacaphony of mechanical gears. It became apparent to me in a split second that it had no intention of stopping. Ever. It had come alive and was angry.
I yanked the plug from the wall socket and ran from the room in terror.
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
If he can make up numbers, then I cam make up words,
this whole thing is utterly stuipfluous.
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.
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
The problem with trying to abstract is that 0 holds no sign. It poses no problem when you multiply with 0, because you don't need to ask about the sign of resulting 0. However, when dividing finite with 0, you know that you have two possible and distant infinite outcomes.
Therefore, if there was 0 and -0, you could claim x/0 = (SIGN(x))*infinity and x/(-0) = -(SIGN(x))*infinity.
Perhaps nullity is used to address exactly this problem of zero's "third sign". There is also similar concept, "infinite complex number", where complex plane is mapped on Riemann's sphere, where south pole is mapped to zero, while north pole is considered "complex infinity". The nullity is "real numbers' only" version of that.
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.
Here's one from the "young whippersnapper" department.
When I was a boy, we programmed air/space craft simultations for NASA.
Not the just abstract videogame types, but actual mechanically-linked 3D motion simulators
that jerked (jerk is a derivative of acceleration, in turn a derivative of velocity, thence a
derivative of position) human test pilots in a shaker cockpit.
Aside: the computer coding involved aviation control math models -> Ratfor -> FORTRAN-> real-time
assembly language -> custom digital I/O in the simulation cockpit, debugged via toggle switch
breakpoints set on a Xerox Sigma 9 console, later supplanted by Foonly machine efforts.
To make a long story short, the aerospace models often attempted divide-by-zero, either from
outright programming bugs or ill-conditioned equations.
So, did we then smash the test pilot into the cabin walls at a high rate-of-change?
No, the intrepid project mechanical engineers, who grokked servo mechanisms and could care less
about snotnose Unix-head punks simply used "mechanical rate limiters" to
overcome and smooth over these "divide-by-zero" disasters.
I'm telling you, even Professor Kahan's IEEE floating-point NAN nomenclature
for calculations didn't save the day for renormalizing these infinities -- how could it,
no self-respecting kernel (Unix or otherwise) has ever executed FP operations, which still
doesn't absolve integer div-zero horrors and concomitant analog duct tape patchwork
to save the day.
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.
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!
you are entirely correct. i believe the proper mod would have been 'enlightening'.
*crickets*
He introduced a multiplicative inverse for the additive identity (0), and added it to the real number field.
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.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
i think it is wrong, given his axioms (as defined here: http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf).
(inf) = 1/0 [A20]
= 1/(-1 * 0) [T77]
= -1 * (1/0) [A13]
= -1 * (inf) [A20]
= -(inf) [A24]
which contradicts his axiomatic supposition of (inf) and -(inf) as unique entities [T41]
lysergically yours
"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."