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.
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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.
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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
Anyone have a link to the Youtube or Gootube version of this?
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 --
He just created a new model, a new rule set, a new abstraction of math to deal with the case of "x/0". In general, dividing by zero is bad for most algorithms. I mean, from a CPU's perspective, I don't see how adding any additional hardware would help.
Only Chuck Norris can divide by zero.
The article and Slashdot's synopsis don't make note of it, but Dr. Anderson isn't claiming to have discovered something new in dividing any number other than zero by itself. The video linked in the article shows him saying that 1/0 = infinity, and -1/0 = -infinity, but 0/0 = capital phi (nullity -- we'll ignore the fact that this usually means the golden ratio in mathematics). Math isn't my area of study so I don't know why 0/0 specifically is so important... the article certainly is very much a fluff piece. Anyone feel like explaining the importance of 0/0?
Wow, since this guy is a computer science prof, maybe he can come up with some value or symbol to represent "nullity." I suggest "NaN" for "not a number." (ducks to avoid rotten tomatoes)
Dividing by zero is not a "problem". It's just IMPOSSIBLE due to the way we structure our species' math. If you want to restructure our math as we know it (which he basically does by inventing his own false reality, so to speak), then you're not solving any problems. You're just being clever, and designing another system.. which has been done hundreds of times.
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.
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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.
In my Graphics class I learned about the Quaternion number field, which is essentially like multidimensional complex (real +imaginary) numbers. In addition to the familiar i, you also have j and k. There is a multiplication table showing what you get when you multiply these things with each other. Why are these useful? Because for some reason or other, they can be used to define 3D rotations "better" than just using two or three angles. And you can make quaternion splines to interpolate between various rotations, allowing you to specify key frames and getting an animation out of it. But it's a really weird sort of number to think about.
yea, actually, you are missing the point.
math is actually the science of making up rules. any real mathematician will tell you that the main idea of math is to start with as few basic axioms as possible, and come up with the rules of the system that follows. see: euclidean geometry, arithmetic. where do the axioms come from? historically, from observing the real world, people saw integers, real numbers, and euclidean geometry. more recently (meaning euclid and a few other clever early dudes, but otherwise in the last 150, maybe 200 years), the axioms are pretty much completely made up. some of them are based on those early systems, integers and real numbers. but there are a multitude of mathematical systems, of all varieties, that have no real world counterpart. and thats what makes it fun.
as for division by zero, it gets us nowhere. the system of arithmetic and real numbers doesn't define division by zero, because that system is used for modeling the real world, where division by zero is meaningless. if you paid attention to the paragraph above, however, you should realize how easy it is to come up with a system where division by zero is clearly defined. my favorite example is the riemann sphere, which can be seen as an extension of the projective real line. of course, in ieee floating point, division by zero is very clearly defined. the result doesn't have a "value" but you can do it, and if you do, your plane doesnt crash.
in short, james anderson is an idiot. yes, i am basing this on my reading of the summary and (pointlessly vacuous) article. if only the video explanation weren't real format...
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"
Anyone feel like explaining the importance of 0/0?
It's what math professors think about when they're too old to bonk a student during those intense one-on-one tutoring sessions.
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.
... where you can actually determine meaningful values for 0/0 in specific cases via calculus?e
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
thegodmovie.com - watch it
... and dividing by zero while on the nullity line lets you go directly to World 9 with only two Warp Whistles!
This fantastic new math is also helpful in solving this intractable problem: http://mcraefamily.com/MathHelp/JokeProofFactoring .htm
How cool is that?
Seriously, it's hard to take someone like this seriously when he uses ignorant scare tactics such as his autopilot example. Either he's performing self aggrandizing hand waving, or he really is completely ignorant about the real world. Trust me - we do account for division by zero in autopilot systems. And - believe it or not - not only does the computer not "stop working" but we actually get a result back. It's called NaN. Furthermore, not only are our systems built with robust libraries that allow us to carry on (no pun intended) we also write downstream code to mitigate propagation of these types of errors. [see Celarier, Sando for a good example of this].
Trust me. This is an inactive account. Regardless of what the
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.
How did this type of crank bullshit get on the BBC ? What's next, an article on the timecube ?!
In Soviet America the banks rob you!
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.
Actually, Mr. L'Hopital pretty much bought his theorem. Rather, Mr. Bernoulli would be the one saying something.
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.
Comment removed based on user account deletion
James Anderson: The numbers all divide by zero. Look, right across the board, zero, zero, zero and...
Marty DiBergi: Oh, I see. And most calculators only go down to 1?
James Anderson: Exactly.
Marty DiBergi: Does that mean it's one smaller? Is it any smaller?
James Anderson: Well, it's one smaller, isn't it? It's not one. You see, most blokes, you know, will be dividing by one. You're on one here, all the way down, all the way down, all the way down, you're on one on your calculator. Where can you go from there? Where?
Marty DiBergi: I don't know.
James Anderson: Nowhere. Exactly. What we do is, if we need that extra push over the cliff, you know what we do?
Marty DiBergi: Divide by zero?
James Anderson: Zero. Exactly. One smaller.
Marty DiBergi: Why don't you just make one smaller and make one be the smallest number and make that a little smaller?
James Anderson: [pause, blank look and snapping chewing gum] These divide by zero.
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 thought he lived circa 500 BC, which would make the problem at least 2500 years old, not 1200, if he were working on it.
... but my RealPlayer divided by zero and crashed.
Claiming to be pedantic on Slashdot is asking for trouble
This has to be a hoax of some kind. I can't believe they let people this dumb teach math.
The same sort of manipulation this guy does can easily be applied to show that 0 = nullity.
0=0^1=0^-1 * 0^2 = 1/0 * 0*0 = 1/0 * 0 = 1/0 * 0/1 = 0/0 = nullity.
How can someone who is supposedly trained and licensed do this to kids.
If you liked this thought maybe you would find my blog nice too:
Is that if I tried this kind of cheating at university, I would have been thrown out of the classroom with a boot-shaped mark on my rear end.
"Discovering" this miraculous new number sounds like winning at the Kobayashi Maru test - by changing the rules of the test itself. Thus, cheating.
Anyone want to attempt a practical application of this so called *invention*?
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.
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.
While a math person would strangle another math person for saying something like that, I was a math/physics major, so I'll tell you that at least in the sciences, you're dead on. It so happens that a lot of really messy operations (particularly trig ones like sines and cosines) over the real numbers look really clean once you realize they are just the real/imaginary parts of simple imaginary functions.
...), which is an infinite sum that only converges to a finite value if the real part of s is greater than 1 (for example, if it's zero, we have zeta(0) = 1+1+1+1+...). We can define its analytic continuation for other values, though, and prove interesting and unintuitive formulae like 1+2+3+4+5+... = -1/12 (which is, amazingly enough, actually somewhat relevant in physics when you look at the Casimir effect or string theory - it's the reason that in bosonic string theory you need 26 dimensions for quantum consistency, as in 2(left/right moving waves)*12(magic number from the zeta formula which counts energies of each mode) = 24, the number of degrees of freedom of a 2 dimensional string world-sheet).
Another way to think of it is that complex numbers are just a really special way of dealing with 2-dimensional geometry, where scaling and rotation are represented by complex multiplication. i corresponds to a 90 degree rotation, which is why i^2 = -1 (i.e. a 180 degree rotation). It's also why you can arbitrarily choose whether i is a clockwise or anticlockwise rotation as long as it's a consistent choice: two -90 degree rotations are equivalent to two positive ones (um...I hate to even bring it up, but that's actually not true in physics, where we have spinors - imagine a book attached to a ribbon which is attached to a table, and imagine turning the book 360 degrees; the ribbon is now twisted, and without further rotation it can't be untwisted, but if you rotate it another 360 degrees, you can undo the twisting without moving the book, by sort of pulling the loop of ribbon over the book - try it out if you're confused. That's the essence of a spinor, that a single full rotation leaves it in the "opposite" state, and that it leaves you confused).
Now I'll take off the science hat and put on the math one...the reason mathematicians love complex numbers is that if you have a function f(z) that is a function of the complex number z = x + iy (where x and y are both real), but not a function of x or y alone (i.e. f(z) = z+z^2+e^iz qualifies, f(z,x,y) = x - y + z does not), there are many subtle and powerful qualities that that function must possess. The one that comes up a lot is that you can do a Taylor expansion of the function and it "works" within a well defined range of values; another nice thing is that integration of the function along closed paths is all but trivial (it's always zero unless it encloses a "pole," i.e. a place where the function blows up in a certain way). As it turns out you can also take a function that you've defined along a single line (or piece of a line) and use its Taylor expansion to extend it to the whole complex plane. This is especially nice for functions like the Riemann zeta function (zeta(s) = 1/1^s + 1/2^s + 1/3^s +
So in summary, complex numbers are very important because they give us so many results that we could not even approach any other way (I haven't even mentioned the more subtle ones, esp. having to do with prime numbers!). To the contrary, the stuff that this professor is pushing seems entirely useless, more of an attempt to push a new term rather than a new concept. Mathematicians have understood infinity and what you can and can't say or do with it for a long time; anything you could even try to explain to a bunch of schoolchildren is either wrong, old news, or irrelevant.
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.
If this is real, who will solve the problem of divide by nullity? Sounds like he's just adding another problem to solve the first one.
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.
AHH! Boas' "Mathematical Methods In The Physical Sciences."
Its a good book. However one of my fav tidbit gleamed from its pages is why Square roots have 2 numbers associated with it and that in actuality the Nth Root of a number has N seprate answers. N-2 imaginary if even and N-1 if odd. Pretty fun stuff.
For a Nth root of a number take 360 degrees of a circle and divide it by N to get a how many degrees between each of the answers for your problem in the complex plane. The hypotinous being the original number and given the fact that you have theta you can find the real and imaginary part of each answer. If you noticed for even Ns the degrees allways land on 180 and 360 refeering to the negative and positive root. So remeber when you take the 8th root of something be sure to check all 8 answers =D
Never could figure out why my girl liked my bitch tits, then I found out she was a lesbian.
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 just solved the P=NP problem. The answer is peeequalsennpeeanswer - a special word I made up which represents a complete proof.
Again:
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<tag/> is a tag that has no content inside it (<tag/> == <tag></tag>).
So, <p/> is an empty paragraph.
factor 966971: 966971
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.
Absolutely. It's also possible to extend the real number system to support something else physicists use all the time, infinitesimals and infinites:s
http://en.wikipedia.org/wiki/Non-standard_analysi
http://en.wikipedia.org/wiki/Hyperreal_numbers
Once you can get your head around ultrafilters, it's really a cool system and, like complex numbers, can allow you to arrive at conclusions that you would have a hard time arrive at without them. But like complex numbers, they don't "really exist". They're just a useful model that helps us solve and understand real-life problems.
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!
Utter rubbish, as usual. Just like those idiotic programmers who start counting from zero.
Repeat after me: Zero is not a number. I didn't hear you, say it again.
Let's get this straight. A number is representative of a quantity.
Zero represent "nothing".
"Nothing" is not a quantity. It is, well...nothing.
Ergo, zero is not representative of a quantity, which means Zero is not a number.
Why is is so hard for people to understand that?
Anyway, math works with numbers, not programmers' fallacious ideas.
It's good that as a rule division by zero is not allowed. Adding this programmers' idea of division by zero would surely add a bug to the system. Yes, and some moron is bound to give us a patch, as this one just did. But guess what, it was wrong in the first place, and should be removed from any support whatsoever.
Have you read my journal today?
I just scanned over his papers. In the second paper he tries to deal L'hopital's rule, lt x-> 0 sinx/x e.g., by saying that we should not consider sinx/x to be continuous at zero. However, we can consider sinx/x to be continuos at 0 for one very good reason - the removable singularities theorem in complex analysis which tells us that in cases like this there is always precisely one function to which sinx/x can be extended so that it is analytic at zero. This theorem guarantees that these are not "harmful extensions" as he calls them but totally harmless extensions. He is a crank. All his idea amounts to is insisting that instead of referring to functions like f(x) = sinx/x as we usually do we would have to call the function f(x) = sinx/x if x!=0, but = 0 if x = 0 - which in light of the removable singularities theorem is unnecessarily clumsy.
Anyway, after reading it i need to sneeze. So should you
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.
Note to ACs: I usually delete AC replies without reading them. If you want to talk to me, log in.
This is a LIGHT BULB JOKE. It may be only slightly funny, and it certainly isn't "insightful", but it's not a troll. It's a JOKE.
No, he didn't prove that f(x)=0/x is continuous. He simply stated that it has a hole discontinuity (which occur when a value of a function is not defined, but a limit exists at that point), not an asymptotic one (which occur when a value of a function is undefined, and a limit is either undefined, positive or negative infinity at the point). There is one other type of discontinuity, a displaced discontinuity. For example, consider the piecewise defined function f(x)={0/x for x != 0, 1, x = 0}. The function is defined at x=0, but its value does not equal the limit at that point.
It has been a nervous year, with people beginning to feel like Christian Scientists with appendicitis.
My calculus class always used to divide by zero... just for very large values of zero.
a/0 is undefined. The end. The limit of a/x as x approaches 0 from above is positive infinity, but the two statements are not the same. Division by 0 breaks a number of rules and can be used to "prove" all sorts of things that aren't true. If 1/0 == inf, then inf * 0 == 1, but any number multiplied by 0 is 0, so inf * 0 == 0, therefor 0 == 1. It just doesn't work. a/0 is undefined. As best as I can tell all this guy is doing is assuming that 1/0 == inf, -1/0 == -inf, and calling 0/0 something else.
-matt
Shouldn't be a problem if you put a space before the exclamation point. :-)
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
Fuck the system? Nah, you might catch something.
I read most of the comments above me, but I didn't see any mention of this. Nullity is a term used in linear algebra to describe the dimension of the null space of a vector space. It isn't as widely used as rank is; however, it still exists.
Complex numbers are useful becuase they are useful in equations and can be used to generate real answers.
I've read his "technical" paper and all it says, in a lot of mathematical jargon, is that once you divide by zero anywhere in an equation the result is 'undefined' only he has now given 'undefined' a new mathematical symbol and a funky name.
Unlike an imaginary number which can give a real single value when used in an equation (e.g. 2i^2+4 = 2) once you divide by zero anywhere in an equation you result can be anywhere in an undefined space between infinity and negative infinity. He calls this space Nullity
So his invention is actually not a mathematical one, it is a gramatic one. Nullity = Undefined, Undefined = Nullity.
Quantum Physics a.k.a. sub-molecular statistics
After years of being a mathematician, I can report that I have yet to see "0" used to mean "nothingness".
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
Yes, that's numberwang!
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."
My apologies if you are receiving multiple copies of this call for papers.
We invite new and innovative submissions for an upcoming symposium to discuss the novel concept of "nullity". "Nullity" was first proposed by Dr. Anderson when he was teaching schoolchildren in 2006A.D. (the actual inventor is still debated). However from that time onwards nullity has been used to prove many phenomenon in everyday life including debt reduction, break ups and even vasectomy. The manuscript should be novel and not published elsewhere. The area of interest includes but is not limited to:
Nullity in network design
Nullity chip design
Evolutionary nullity
Educating children on nullity
Nullity based algorithms
Please submit the above papers directly to Dr. Anderson at an.ders.on@__.__ (Please install the nullity plugging to display email address). The symposium will be held from 29-35 March 300G.E. on First Foundation.