Division by Zero (Please, no new threads about this)

[Architect]

First off, I'd like to say that I recall there being a math section here when I joined. When I went to post this- no math section... so I just decided to put it here. Sorry if it's in the wrong place.

Anyway.. I'm a high school student, so my knowledge is limited.

I've been thinking about this:

0/0 = ?

The way I was taught fractions was that 0 / anything = 0.
However, I was also taught that anything / 0 = undefined/no solution/whatever.
And of course I was taught that anything over itself is 1.

So I was wondering, since all the people here seem to be of a higher intelligence level than I, which one of those is the actual solution to 0/0?

[Jesse]

nullinity.

[Narsil]

Just don't type it into your calculator.

It may explode.

[Alisto]

Doesn't exist/undefined. Here's proof:

1x0 = 2x0

0/0 = 1/2 = 2/1 Obviously not possible if 0/0 = 1 or 0.

[SargeZT]

It's undefined (read: doesn't exist), but the lim x->0 of x/x is 1, but that isn't what it equals. It really equals nothing.

Or Nullity. Let's go with that.

[Gelsamel]

Undefined Architect.

[damienthebloody]

Or Nullity. Let's go with that.

if you use that word again, i'm going to get very upset. you wouldn't like me when i'm upset.

[Gelsamel]

Do you turn into some crazy demon?

[Shadowfish]

This forum is amazing. I saw this topic on another forum, and it took 5 pages of fierce debate before the people who though 0/0=1 gave up.

[3.14159265...]

0/0 = God*Black holes

God= No such thing = Nothing
Black holes= Where god divided by zero= /0

The contradiction is from what others have noted above

[bbctol]

the people who though 0/0=1

Ah... interesting stupid theory. Who are said people? How do they do something so... clearly wrong?

[Gelsamel]

This forum is amazing. I saw this topic on another forum, and it took 5 pages of fierce debate before the people who though 0/0=1 gave up.

Just don't post a 0.999... = 1 topic on that forum, and you'll be fine.

[Shadowfish]

Just don't post a 0.999... = 1 topic

Lol! That one was painful.

Who are said people?

The megatokyo fora. I hung out there a few years ago. I wish I could remember the arguments.

[EvanED]

Has anyone opened the Monty Hall can of worms there?

[SpitValve]

Several times.


0.999...=1 has been done several times too.


And there was something about infinity a while back too...

[mattmacf]

So I was wondering, since all the people here seem to be of a higher intelligence level than I, which one of those is the actual solution to 0/0?

The problem with evaluating a statement as simple as 0/0 is that we don't know how big or how small the zeros are relative to each other (I'm not kidding here!) The idea here is that 0/0 is in indeterminate form. Basically, saying 0/0 isn't really enough information to get an answer (just saying this gives us an undefined answer). However, (you'll probably learn this if you take a calculus class) if 0/0 is the answer you get when evaluating a particular expression, there may indeed exist a definite limit that the function converges to. To find it we generally use L'Hopital's(sp?) rule to take the derivative of both the numerator and the denominator and then evaluate the result.

I believe a previous poster mentioned lim(x -> 0) [x/x]. Simply "plugging-in" 0 for x gives us 0/0. Using L'Hopital's rule and taking the derivative of top and bottom gives us lim(x -> 0) [1/1] which is indeed 1. A caveat, however, is that just as 0/0 can equal 1, it can also equal 0, infinity, 42, or just about any other value. That's why we call 0/0 an indeterminate form. There are also a variety of others too, such as infinity/infinity or 0^infinity (check the wikipedia article for the others). This, my friend, is what makes calculus (well math in general) so much fun

[Aoeniac]

0/0=more work.

[Owijad]

This forum is amazing. I saw this topic on another forum, and it took 5 pages of fierce debate before the people who though 0/0=1 gave up.

I'm perfectly willing to fabricate controversy if you'd like.

[Jesse]

0/0 definitely equals 1 because God says so.

Lolol.

[Akira]

This forum is amazing. I saw this topic on another forum, and it took 5 pages of fierce debate before the people who though 0/0=1 gave up.

I'm perfectly willing to fabricate controversy if you'd like.

Fabrication of controversy is all well and good, but it appears as though the logical arguments have been presented. Further discussion is simply going to be rehashing the equation and re-wording the argument.

Nothing will be accomplished.

Except I'll get to put off going to bed later and later until I can't wake up for school tomorrow morning.

[thefiddler]

nullinity.

That article was huge when I first started posting regularly. It kind of made me cry a little inside. But that is what I thought of when I saw the topic title. >< Me = lame. (There are no proofs to disprove that. )

[Alisto]

The megatokyo fora. I hung out there a few years ago. I wish I could remember the arguments.

Enough said. I think I hung out there at the same time.

Indeterminate form stuff

Way to ruin the party. Besides, indeterminate forms only apply to limits. Algebraically, which is what it seems like he was asking about, it's simply undefined. :p

[Sitnaltax]

There is no solution. The expression is undefined if we are talking about real numbers.

I became quite angry when I read that article where the smug bastard introduced his made-up word "nullity" like he was some kind of brilliant mathematical genius. He was clearly trying to define a new set of numbers with not even a scratch of the theory required for such an undertaking, and his ignorance was painful.

There's a good article that pretty much showcases his incompetence here:
http://en.wikinews.org/wiki/British_com ... ematicians

(No, I'm not a mathematician. I gave up on that when I realized that "only minor difficulties understanding integral calculus" means "nowhere near good enough to pursue serious mathematics.")

[lanicita]

I think it's more fun as a word problem: Divide nothing into no groups. Oh look, you're done!

I think that in that case, the answer is "zen." (I'm referencing Ozy and Millie if nobody got that )

[bitwiseshiftleft]

Shoot, you can just type this into your favorite programming language or math tool...

Prelude> 0/0
NaN

See? It's Not a Number!

Amusingly, Google just does a normal web search if you enter a calculation that would divide by zero. This is true for other undefined things too, like csc(pi). Google's fine telling you what csc(pi/2) is...

[Cosmologicon]

The problem with evaluating a statement as simple as 0/0 is that we don't know how big or how small the zeros are relative to each other (I'm not kidding here!) The idea here is that 0/0 is in indeterminate form. Basically, saying 0/0 isn't really enough information to get an answer (just saying this gives us an undefined answer).

Be that as it may, 0^0 is an indeterminate form that we usually define to be 1, with good reason.

[parallax]

The division operator is not defined when the divisor is zero. And try not to be tricked by 1/0 = infinity. It's not true. That's just shorthand for saying "If the number you're dividing by gets smaller and smaller, then the quotient will get bigger and bigger."

"0/0" is a meaningless term. It has about as much mathematical meaning as "2 + apple".

[skeptical scientist]

The problem with evaluating a statement as simple as 0/0 is that we don't know how big or how small the zeros are relative to each other (I'm not kidding here!) The idea here is that 0/0 is in indeterminate form. Basically, saying 0/0 isn't really enough information to get an answer (just saying this gives us an undefined answer).

Be that as it may, 0^0 is an indeterminate form that we usually define to be 1, with good reason.

I don't know if I agree with this. It's an indeterminate form in the context of limits, and it's defined to be 1 in the context of the empty product, which are two very different situations (one is looking at the limit where the exponent is a very small number, and one is only assigning meaning to integer exponents). Either it is an indeterminate form, or it is defined to be 1, but not both.

[Cosmologicon]

0^0 is an indeterminate form that we usually define to be 1, with good reason.

I don't know if I agree with this. It's an indeterminate form in the context of limits, and it's defined to be 1 in the context of the empty product, which are two very different situations (one is looking at the limit where the exponent is a very small number, and one is only assigning meaning to integer exponents).

No, evaluating the empty product isn't the only time it's handy to define 0^0 = 1. The most obvious is the binomial theorem, which actually works fine for non-integer exponents, even though this fact isn't usually mentioned. It comes up in calculus too. For instance, how would you write the power series for exp(x) in sigma notation? Most people would answer Sum x^n/n!, n = 0...infinity, rather than 1 + Sum x^n/n!, n = 1...infinity.

Of course, defining 0^0 doesn't let you do anything that you can't do in a more cumbersome way by not defining it, but the same is true with 0!, sqrt(-1), and even 2 + 3.

[parallax]

"0^0=1" is a perfectly valid statement and it's the usual definition because it remains consistent with most of number theory. It does require that you tiptoe around some of the usual properties of ^ as statements like "0^0 = 0^1 * 0^-1" are obviously meaningless, but it's not any worse than "1/0 * 0 = 1".

0^0 is also considered an indeterminate form in calculus because the function x^y is not continuous at (0,0). Thus, when evaluating the limit "lim (x,y)->(0,0) x^y", it is incorrect to say that the limit is 0^0 or 1. The term indeterminate form should only be used when refering to limits.

So, 0^0 can be defined to be 1 and be an indeterminate form at the same time.

[skeptical scientist]

For instance, how would you write the power series for exp(x) in sigma notation? Most people would answer Sum x^n/n!, n = 0...infinity, rather than 1 + Sum x^n/n!, n = 1...infinity.

Hmm, that's a good point. Although I would probably just write 1+x+x^2/2+x^3/3!+.... I've never really thought about that before. I guess I naturally just remove removable discontinuities. For example, if f(x)=e^(-1/x^2), then for me f(0)=0, although strictly speaking it should be undefined unless you've defined it using a piecewise construction. But it seems to me that when you're defining 0^0=1 in that sum you're not doing it because an indeterminant form is 1, you're doing it because this particular instance of the indeterminant form arises from taking the limit of x^0 as x approaches 0, which is obviously 1. You're not saying "we define the indeterminant form 0^0 to be 1", which would be silly, since it's an indeterminant form.

I think I was confused because it seemed you were saying that "in most cases we define all instances of this indeterminant form to be 1", but what you were really saying was "in most cases we define this indeterminant form to take on its limit, which is usually 1 when it arises", which is true since if you look at the limit of f(x)^g(x) as x tends to x_0, and f(x_0)=g(x_0)=0, you will get 1 if f and g are analytic, and f is not identically 0, which is usually the case in practice.

[TheTankengine]

0/0 definitely equals 1 because God says so.

Lolol.

Srsly. i liek readz it n teh bIb7ezorzorz!!!one-diddy-onedy!!

And God totally said: Zero divided by zero shall be One.
And it was teh Good.

[Alisto]

0 = no, 1 = yes
No = negative, 1 = positive
Therefore 0 = negative, 1 = positive
negative/negative = positive
0/0 = 1
QED

[3.14159265...]

I created a very similar yet very profanitic response, I will not post it.

[Puellus Peregrinus]

Oh gee 0/0 is clearly 0.01

1=100%
1/100=%
%=0.01


I also like the analogy "There are 0 apples. The apples are divided among 0 people. How many apples will each get?"

[Yakk]

"There are 0 apples. The apples are divided among 0 people. How many apples will each get?"

3, but one will be bruised.

[Hix]

Another reason that 0^0 is usually taken to be 1 is so that we can have:
|A^B|=|A|^|B| for any sets A and B (A^B is defined as the set of all functions from B to A),
since there is exactly one function from {} to {}.

As for 0/0, I wonder why I've never thought of the apple analogy... it's good! If one really thinks about it, it shows why the answer can be anything: How many apples will each get? Any number, of course! There are no people, so it is vacuously true that each of them gets, say, 5 apples.

Usually, I resort to an argument along the lines of: Start counting multiples of 0 aloud (starting with 1 times 0). You may choose to stop counting, but only if the last number you said was "0". Suppose you've stopped counting. How many multiples have you counted?

[cmacis]

Not defined, but you can get some interesting nonsense by playing around with division by zero.

Don't be too hard on those who try; some of the greats used 1/0=infinity so infinity/0=1.

[Strilanc]

One way to 'divide by zero' is redefine arithmetic to work on sets of numbers.
{1} + {2} = {3}
{1,2} + {3,4} = {4,5,6}
etc.

So {1}/{0} = {}, {0} / {0} = R, {} / {} = {}

But of course this is nothing like normal arithmetic and we've just avoided the problem by changing the system we work with. For an example of how this is not like normal arithmetic: there exists an X such that for all A, A + X = X (similar to 0*A = 0): X = the empty set. {} + A = {}. Also, A / B * B might not be A: ({1} / {}) * {} = {}, ({1} / {0}) * {0} = {}, ({1}*{0}) / {0} = R.

[Ansain]

I dont think 0/0 can equal one I think it just stays as 0 just like any other number times 0. I mean think about if I have 0 apples and divide them ammongst 0 people do they each get 1? if you assume N/0 = infinity or N/x approaches infinity as x approaches 0. Since the rate at which it approaches infinity is proportional to N you can say N/0 = N times infinity. Then you can simplify 0/0 to zero times infinity.

The way I see that is each block is infinitly large and goes on forever. However you dont have any blocks so you still dont have anything. Or each box contains absolutely nothing and you may have an infinite number of them, but there is still nothing there.

so 0/0 = 0 if you can agree that N/0 = N times infinity.