Divide by zero, solved?

Ace Pace · Post by **Ace Pace** » 2006-12-07 09:50am

Abit confusing, and I can't see the video link.

1200-year-old problem 'easy'
Schoolchildren in Caversham have become the first in the country to learn about a new number - 'nullity' - which solves maths problems neither Newton nor Pythagoras could conquer.

Dr James Anderson, from the University of Reading's computer science department, says his new theorem solves an extremely important problem - the problem of nothing.

"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."Watch a video report from BBC South Today's Ben Moore, then let Dr Anderson talk you through his theory in simple steps on the whiteboard:

Computers simply cannot divide by zero. Try it on your calculator and you'll get an error message.

But Dr Anderson has come up with a theory that proposes a new number - 'nullity' - which sits outside the conventional number line (stretching from negative infinity, through zero, to positive infinity).
'Quite cool'

The theory of nullity is set to make all kinds of sums possible that, previously, scientists and computers couldn't work around.

"We've just solved a problem that hasn't been solved for twelve hundred years - and it's that easy," proclaims Dr Anderson having demonstrated his solution on a whiteboard at Highdown School, in Emmer Green.

"It was confusing at first, but I think I've got it. Just about," said one pupil.

"We're the first schoolkids to be able to do it - that's quite cool," added another.

Despite being a problem tackled by the famous mathematicians Newton and Pythagoras without success, it seems the Year 10 children at Highdown now know their nullity.

Zac Naloen · Post by **Zac Naloen** » 2006-12-07 09:53am

High down?

eugh, I'm shocked they didn't use Maiden Erleigh, that school has better Ofsted reports and is about 5 minutes from University of Reading campus.

Kuroneko · Post by **Kuroneko** » 2006-12-07 10:18am

This is not mathematics. He "solves" the problem of 0^0 by arguing that this is 0^(1-1), and in his system it comes out to be his new nullity Φ. Repeating his exact steps on 0^0 = 0^(2-2) gives Φ² = Φ, so it is idempotent. Furthermore, by rearranging terms, we have nΦ = n0^0 = Φ. This clearly breaks the field structure of the reals for no good reason, thus breaking algebra. The real issue is this: Φ becomes a solution to every polynomial equation with no constant term.

It's not even proper computer science. Dr. Anderson has just discovered the NaN (non-a-number) symbol of IEEE floating point and called it "nullity." This is newsworthy? My respect for BBC has just died.

phongn · Post by **phongn** » 2006-12-07 10:20am

Kuroneko wrote:It's not even proper computer science. Dr. Anderson's has just discovered the NaN (non-a-number) symbol of IEEE floating point and called it "nullity." This is newsworthy? My respect for BBC has just died.

That, and he makes it out to be a much bigger problem than it should be - any competent software engineer in a critical field will check for divide-by-zero errors (or divide-by-close-to-zero)

Mad · Post by **Mad** » 2006-12-07 10:39am

Kuroneko wrote:It's not even proper computer science. Dr. Anderson has just discovered the NaN (non-a-number) symbol of IEEE floating point and called it "nullity." This is newsworthy? My respect for BBC has just died.

That was my first thought as well. "So, he's just mapping NaN to his new 'nullity'?"

This doesn't "solve" anything that I can tell, because coding a new system to handle "nullity" in the case of a divide by zero would be the same as coding it to handle NaN. A computer program, after all, isn't going to suddenly be able to properly add 5 to "nullity" anymore than it can add 5 to NaN.

As for how much it breaks... using GCC 3.3.2, I just did 5 / 0 and added 5 to the result. I got "Inf" both times. Perhaps not the correct answer, but it hardly crashed.

haard · Post by **haard** » 2006-12-07 10:39am

Kuroneko wrote: It's not even proper computer science. Dr. Anderson has just discovered the NaN (non-a-number) symbol of IEEE floating point and called it "nullity." This is newsworthy?

My thought exactly while reading this. NaN has been around since... forever? *checks* IEEE 754 is from 1985, which for me ~= forever (first used a computer -89)

phongn · Post by **phongn** » 2006-12-07 10:51am

Mad wrote:As for how much it breaks... using GCC 3.3.2, I just did 5 / 0 and added 5 to the result. I got "Inf" both times. Perhaps not the correct answer, but it hardly crashed.

Isn't the behavior of divide-by-zero undefined in the C standards?

Admiral Valdemar · Post by **Admiral Valdemar** » 2006-12-07 11:10am

So really, as I was expecting upon reading the title, he's redefined something and said "Hah! I can do this now". Gee, I bet I could divide by zero if I maintained that "zero" was actually "two".

EDIT: Additionally, it should be noted that plenty of people who thought this through posted their comments on the page linked. It is also telling of the lowering of journalistic standards within the BBC who have so far managed to report on pseudoscience, kill the award winning series Horizon with stupid dumbed down stories and make the most basic errors on stories. It's like they don't care anymore. At least New Scientist, who supply a lot of the stories the press later pick up on, doesn't actively promote dodgy ideas, but just lists articles dealing with them.

Course, anyone who wants real science should be reading Nature, but you have to actually think about that.

Oh, and maths isn't science, ergo, sucks.

Mad · Post by **Mad** » 2006-12-07 11:13am

phongn wrote:
Mad wrote:As for how much it breaks... using GCC 3.3.2, I just did 5 / 0 and added 5 to the result. I got "Inf" both times. Perhaps not the correct answer, but it hardly crashed.
Isn't the behavior of divide-by-zero undefined in the C standards?

I don't know, but the GCC-compiled (C++) program did die when I used integers instead of floating point numbers. The floating point numbers would have to follow IEEE standards, integer divide by zeros may not be handled consistently.

In any case, with the IEEE floating point numbers, dividing by zero isn't an unsolved problem. (Granted, the program may not work correctly when using the computed values, but that problem will exist if using this "nullity" thing, too, so nothing new is solved.)

Velthuijsen · Post by **Velthuijsen** » 2006-12-07 11:14am

Not sure about C, a quick trawl through the net gave a reference to the 9X revision of C incorperating IEEE 754, labelled iec 559 the same standard just another standards body that put it's label on it.
For C++ it is in the standard.
And that means (if the IEEE 754 reference I have used for my work is still correct) that division by zero is a special NaN, special enough in any case to merit it's own designation. Instead of the general undefined code of NaN that is.

Post by **Darth Wong** » 2006-12-07 11:24am

There is no way to "solve" the problem of the concept of division simply not being compatible with a denominator of zero. Division by zero will always be an undefined result. Assigning a name to that undefined result won't change anything.

Zac Naloen · Post by **Zac Naloen** » 2006-12-07 11:28am

Out of curiosity under what situation is something like a pace-maker ever going to divide by zero anyway?

Admiral Valdemar · Post by **Admiral Valdemar** » 2006-12-07 11:30am

Zac Naloen wrote:Out of curiosity under what situation is something like a pace-maker ever going to divide by zero anyway?

There isn't any. This is addressed in the first posted comment on that page. This guy is also not affiliated with Reading's maths dept., so anything he says is purely from the computing dept. And it would seem a great many people are trying to retcon this story now.

Dooey Jo · Post by **Dooey Jo** » 2006-12-07 01:01pm

Mad wrote:
phongn wrote:Isn't the behavior of divide-by-zero undefined in the C standards?
I don't know, but the GCC-compiled (C++) program did die when I used integers instead of floating point numbers. The floating point numbers would have to follow IEEE standards, integer divide by zeros may not be handled consistently.

Try checking for exceptions. I'm pretty sure it's C++ standard that an integer division by zero should throw an exception.

Ariphaos · Post by **Ariphaos** » 2006-12-07 01:24pm

phongn wrote:Isn't the behavior of divide-by-zero undefined in the C standards?

It's defined as undefined - ie, NaN. My beef on Slashdot was along similar lines - "He redefines undefined (which is high school math), gives it a new name and calls it new math? BS."

Darth Servo · Post by **Darth Servo** » 2006-12-07 01:50pm

If dividing by zero equals this "nullity", the following multiplication equations are valid.

"nullity" times zero = 5
"nullity" times zero = 253
"nullity" times zero = 190.44

<repeat ad nauseum for every real number>

Post by **NecronLord** » 2006-12-07 01:54pm

In other news, 20/1 is a new number, Equivalncey. To be represented by the symbol ¤.

Wait, do I need to get a doctorate in mathematics before I can come up with bullcrap names for numbers?

EDIT: What's more, how does he propose that this will make operations with 1/0 viable for computers? Given that such an operation is either some flavour of ∞ or as computers preffer 'take a hike bozo, I'm not even going to try.' It seems a lot like he's just assigned a new name to a n/0 error and is saying it's a breakthrough. How are we to get a computer that can process an n/0 command and continue to work based on Ф, any more than on NaN?

Prozac the Robert · Post by **Prozac the Robert** » 2006-12-07 02:24pm

The excellent www.badscience.net has a link to his paper.

It starts out odd, and then sorta remains that way.

Section 2. Axiomatisation. In 2.1 he has (amongst other things) defined some properties of nullity:

Phi + a = Phi

-Phi = Phi

Phi.a = Phi

1 / Phi = Phi

These seem to ensure that once you get Phi in a calculation you get stuck with it. Not very promising.

Of course, the worst thing in the section is this one:

0^-1 = infinity

At least he's done us the courtesy of listing it as an axiom, since it means that it's the only thing that needs disproving.

3.1 is pointless since it just shows a bunch of machine proofs. 3.2 shows some maths that people ought to be able to follow, but I can't be bothered. After all, the axioms are stupid. If anyone can be bothered, 4 and 5 are ramblings. I have too much stuff I'm supposed to be reading for my course to want to read them. If there is any point to his stuff, it's in 3.2 or 4. I doubt it exists though.

phongn · Post by **phongn** » 2006-12-07 02:32pm

Xeriar wrote:It's defined as undefined - ie, NaN. My beef on Slashdot was along similar lines - "He redefines undefined (which is high school math), gives it a new name and calls it new math? BS."

No, if x/0 = NaN in C, then the behavior is defined even if the result is effectively undefined. I thought that C simply didn't define the behavior of divide-by-zero entirely and thus left it up to the compiler designers.

Dooey Jo wrote:Try checking for exceptions. I'm pretty sure it's C++ standard that an integer division by zero should throw an exception.

C++ will not throw an exception

Ariphaos · Post by **Ariphaos** » 2006-12-07 03:28pm

phongn wrote:
Xeriar wrote:It's defined as undefined - ie, NaN. My beef on Slashdot was along similar lines - "He redefines undefined (which is high school math), gives it a new name and calls it new math? BS."
No, if x/0 = NaN in C, then the behavior is defined even if the result is effectively undefined. I thought that C simply didn't define the behavior of divide-by-zero entirely and thus left it up to the compiler designers.

Well there are a pair of overflow flags in the status register (in x86 processors). When the processor does a divide by zero, one or both of these is triggered (I think both, but I forget the specifics... my assembly courses were years ago - there are two flags representing different types of overflow situations).

The specific return from the processor depends on various set flags, but the condition is considered undefined and it's up to the flags and or the compiler and programmer-set directives to define what the result should be - some use Max_Float or Max_Int instead of NaN, for example - especially since the NaN convention is fairly recent (my oldest C books make no mention of it whatsoever despite discussing divide by 0 errors).

Borland's early compilers crashed on error if it wasn't specifically handled - I think Div0 was #200 (I saw it a lot >_>)

B5B7 · Post by **B5B7** » 2006-12-07 10:36pm

Computers simply cannot divide by zero. Try it on your calculator and you'll get an error message.

One can get an error message with all sorts of things on a calculator.
Division by zero isn't simply an artifact of limits of calculator or computer operations - it is fundamental axiom of maths that can not divide by zero.

bilateralrope · Post by **bilateralrope** » 2006-12-07 11:16pm

Computers simply cannot divide by zero. Try it on your calculator and you'll get an error message.

Most can't calculate the square root of negative 1 either, but complex numbers do have their uses.

So what problems does this definition solve, that defining x/0 = infinity doesn't ?

DPDarkPrimus · Post by **DPDarkPrimus** » 2006-12-07 11:26pm

[quote]
Dear Dr. Science,

Why can't you divide by zero?

I can and often do divide by zero, but only after I've made the necessary preparations. First of all, I fast for 48 hours, consuming during that time only mildly fluoridated water. Next I don my special Teflon division-by-zero suit. Then I put on my digitally recorded compact disc of Gregorian chants and begin with dividing very small numbers by other very small numbers. As the numbers get smaller, the sparks begin to fly. If all goes well, I take a deep breath and divide a very small number by zero. There's a flash of light, a muffled roar, and when I regain consciousness, the lab is filled with smoke and the scent of burning mylar. So, you see, you can by divide by zero if you really want to. Chances are....you just don't want to badly enough.

Sriad · Post by **Sriad** » 2006-12-07 11:35pm

Darth Wong wrote:There is no way to "solve" the problem of the concept of division simply not being compatible with a denominator of zero. Division by zero will always be an undefined result. Assigning a name to that undefined result won't change anything.

Yea. L'hopital's rules let you do everything you CAN do in terms of meaningful division by 0 already.

Lord Zentei · Post by **Lord Zentei** » 2006-12-07 11:35pm

BBC really needs stricter editorial standards for their mathematical and scientific news articles. And by that I don't mean some random geek, I mean they need actual professionals with credentials on their payroll.

Despite this, they are markedly less bad than the great majority of mainstream media - a fact which is all the more shocking. I still cringe at the recollection of a news article on CNN.com which discussed a paper on abiogenesis where it was postulated that certain types of clay could have formed the protective environment for early self-replicating molecules. CNN's take: "Science now shows that life was created from clay, just as faiths preach". Ugh.