Issues with 0, and why it is dangerous.

[Xannari Ferrows]

So recently I've been asked by Nic about a few conundrums with 0, and why it is a strange thing. I was even asked if it was a real number!

The 3 primary questions that I wanted to bring up here were:

1. What is the true answer for something divided 0?

2. 0⁰ has some bizarre things about it. Calculators are told that anything⁰ is 1, and is told to stop the operation there. However, is that really what the math is doing?

3. In fact, what about 0/0? I mean, anything over itself is equal to 1, but then we are back to question 1.

So it should be clear that 0 is a thing to be handled with care. Well, let's go ahead and try to answer some of these questions, and I'll leave everyone above to interject whatever they want.

"What is the true answer for something divided by 0."

This is a pretty interesting question to bring up. Amateur mathematicians would assume infinite, because division is nothing bit hyped subtraction.

For example, take 100/10. How many times do you take 10 from 100 to get to 0? Obviously your answer is 10 times.

So that's all well and good, but what about 100/0? Prepare for the brainf%ck:

So, take away 0 from 100 as many times as it takes to get to 0. Okay, take 0 away, we still have 100. Do it again... 100. ...One more time... 100... This is getting nowhere.

This is a problem, because you would think that the amount of times to get to 0 never ends, and is therefore infinity. But it's slightly more complicated than that.

Infinite is not a number, and you wont treat it like a number. It's an idea. You can't say something is equal to infinity.

To demonstrate, let's cheat a little and say 1/0 = ∞ Now this would be all fine and good, but now answer 2/0. It is also infinity, right? So... 1 = 2? That makes no sense to me.

This is why you can't have anything equal to infinity. It just leads to a whole lot of nonsense.

This may be a bit confusing to follow, so here's another demonstration. If you get everything, just skip it.

We can imagine this in another way. Imagine the fraction 1/X. As X approaches 0, your answer is closer and closer to infinity. Therefore you can conclude that your answer is infinity, right? Well, it's a bit different than that. It depends on how you approach 0. Say for instance, you approach from the negative side. Your answer would change to -∞ assuming you're cheating and say that approaching form the positive side equals +∞.

Now, you may think that positive and negative infinity are infinitely different, and that's cool, but use caution when going around saying that. Some people might get a little mad at you.

In conclusion, dividing by 0 is what is known as an undefined variable. In more complex numbers, you could say I.

But to wrap it up here, I'm willing to stand by dividing by 0 is a big ball of doom filled nonsense that should never be discussed.

"0⁰ has some bizarre things about it. Calculators are told that anything⁰ is 1, and is told to stop the operation there. However, is that really what the math is doing?"

This has been debated for a long time. Here are the laws:

0 to the power of anything is 0.

Anything to the power of 0 is 1.

So what happens when these 2 collide? Obviously plugging into your calculator gives you 1 and you could call it a day there, but lets look a bit deeper.

Calculators are programmed to display 1 whenever something is raised to 0, and told not to give it any thought. Just display the number and move on. However, it's bigger than that.

Now, imagine a calculator without this feature. What would it give you? 0? 1?

Imagine the problem X^x. You want your limit as 0, and to approach it from both sides.

As you approach 0 from the positive side, you end with +1. Now, as you approach from the negative side, you also end up with +1. So we could stop here, but I can tell you that it's more complicated.

This is using just a real number line, but what about complex numbers? Draw in your imaginary plane, and now there are tons of ways to approach the origin along this plane. For these approaches, your limit is different and you don't get 1 anymore.

It all falls apart in the complex numbers, and this is why mathematicians have marked this as undefined. The limits vary.

"In fact, what about 0/0? I mean, anything over itself is equal to 1, but then we are back to question 1."

Finally. Something manageable. And no, we're not back to question 1.

0/0 is a bit of an issue because it doesn't equal one thing.

Imagine a number line. What are the points at the origin? (0,0) So, x/y is 0/0. This is bad news. Is it 0? Infinity? What is it exactly?

Well, in truth, it can be anything. 0/0 follows the laws that anything over itself equals 1, so this applies to 2/2, 3/3, 4/4, ect. This means that 0/0 equals basically everything that can be put over itself and equal 1.

Now, this does conflict with what I said earlier. Nothing can be divided by 0 because it would lead to infinity and start causing paradoxes.

While this may be true, it is not the same situation as this, exactly, because, while infinity may not be equal to itself, 0 is.

0/0 equals anything that can equal 1 when put over itself. So basically, everything.

In conclusion, 0 is a dangerous thing. There are a lot of problems, and maybe you should try to work out those problems. After all, this stuff is terribly fun.

But I'll be done for now. Like I said earlier, if you have anything to interject, feel free to do so above and I'll get back to you as soon as I can.

Bye for now!

Xannari

Edited January 29, 2015 by Xannari Ferrows

[Lohan2008]

Nothing divided by Nothing equals Nothing. 0/0= 0

[Xannari Ferrows]

Nothing divided by Nothing equals Nothing. 0/0= 0

This was something I actually saw earlier when doing a bit of research. This exact same comment, and I'll tell you why it isn't true.

Nothing divided by nothing equals nothing, and nothing is equal to itself, so your end ratio is 1x, not 0x. This means that 0/0 cannot equal 0.

This of course is an issue, because the math implies that nothing divided by itself equals everything but nothing. So the answer has gone unexplained.

[kerbiloid]

http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule

[magnemoe]

1/0 is infinite because 1 divided by an very small number becomes an very large one. 1/0.001=1000, 1/10^-10=10^10 and so on.

However we could just agree that x/0 is 0, this would give adequate results for most cases then you run into it in computer programs rather than raising an error.

[Z-Man]

Sorry, but you're talking mostly gibberish.

1. What is the true answer for something divided 0?

In the realm of Real or Complex or Integer or Natural numbers, the answer is simply that it is not a legal operation, and that's the end of it. Yes, you can add infinity to your realm of numbers to "solve" this problem and make division a globally defined continuous operation, but you inevitably run into problems with the other axioms you want numbers to obey.

And yes, twice infinity equals once infinity. Look up the Hilbert Hotel.

2. 0⁰ has some bizarre things about it. Calculators are told that anything⁰ is 1, and is told to stop the operation there. However, is that really what the math is doing?

Yep. Your rule "0 to the power of anything is 0" is not an actual rule. For Natural numbers (0 included) a and b, the only rules are:

1. a⁰=1 for all a

2. a^b=a*a^b-1 for b > 0 and all a.

For integer/whole numbers, you extend rule 2 to all b and you're done. So no argument there, according to the real rules, the only valid value for 0⁰ is 1 in those domains.

Now, for Real numbers, if you want, you can define a^b as a completely new function and depending on how you do it, it can be consistent to have 0⁰ equal either zero or one. However, if you operate on real numbers and a result of your calculation is 0⁰, that means you have at some point not done a proper limit process; and depending on what that proper limit process would be, the true answer you seek could be anything (a^b is only continuous and well defined for all b if a > 0), so the value you assign to 0⁰ is somewhat arbitrary and of little significance. Usually, one would pick 1 for consistency, then.

3. In fact, what about 0/0? I mean, anything over itself is equal to 1, but then we are back to question 1.

See question 1. Not legal. Possible to define away, but you get other problems. If your calculation result is 0/0, then you again forgot a proper limit process, in which case I refer you to kerbiloid's answer.

[N_las]

Zero is not a real number, but just an idea? Any number is just an idea. And what Z-Man said. It is all defined pretty well and consistent in mathematics. We had hundreds of years to think about all this.

[Mast]

Why is everyone so intent on making sense of illegal operations?

Assume you have a cake. You divide the cake over 2 people. Everyone gets halve of the cake.

Assume you have another cake. You divide the cake over 0 people. Nobody gets cake, but somehow we still assume it's divided.

Yea, right.

The one and only exception to being able to divide by zero is the rule of l'Hôpital, which is indicated by kerbiloid and has it's own quirks (there are situations where it is not valid, as with many mathematical rules).

Do note non-positional numeric systems like the Romans used don't have a zero. They had no use for it.

[ZetaX]

Infinite is not a number, and you wont treat it like a number. It's an idea. You can't say something is equal to infinity.

To demonstrate, let's cheat a little and say 1/0 = ∞ Now this would be all fine and good, but now answer 2/0. Is is also infinity, right? So... 1 = 2? That makes no sense to me.

This is why you can't have anything equal to infinity. It just leads to a whole lot of nonsense.

No nonsense in infinity. All you wrote there could be read as elements from 1-dimensional projective space, which for the sake of this post shall be defined as fractions a/b, where we don't allow a,b to both be 0 and consider a/b and c/d "the same" if ad=bc.

Now for b not 0, the term a/b is just an ordinary number. And all terms of type a/0 are "the same", call that infinity. So surely 1/0 = 2/0, but you can't get 1=2 from that.

All the above is just definition and convention, thus so far no further rules or ways to calculate are given, but I will leave it at that.

0 to the power of anything is 0.

That's wrong. Only to a _positive_ power it is. 0^(-1) would be the same as 1/0 and we already had that...

Anything to the power of 0 is 1.

Indeed and that's the one that works well.

So what happens when these 2 collide? Obviously plugging into your calculator gives you 1 and you could call it a day there, but lets look a bit deeper.

Calculators are programmed to display 1 whenever something is raised to 0, and told not to give it any thought. Just display the number and move on. However, it's bigger than that.

Now, imagine a calculator without this feature. What would it give you? 0? 1?

Imagine the problem X^x. You want your limit as 0, and to approach it from both sides.

As you approach 0 from the positive side, you end with +1. Now, as you approach from the negative side, you also end up with +1. So we could stop here, but I can tell you that it's more complicated.

You can't argue by limits that way. The property of continuity, i.e. lim f(x) = f(lim x), is not something that is automatically satisfied. Indeed: in this case, it cannot be satisfied as already argued by yourself.

This is using just a real number line, but what about complex numbers? Draw in your imaginary plane, and now there are tons of ways to approach the origin along this plane. For these approaches, you limit is different and you don't get 1 anymore.

This has a much greater flaw than 0^0: you need to define complex exponentiation. Which is messy and best not done at all unless you talk about multi-sheeted covers.

It all falls apart in the complex numbers, and this is why mathematicians have marked this as undefined.

It is not left undefined by most mathematicians. 0^0 = 1 is an ubiquitous definitions that works well with everything you would need it for.

a) Sets: If A is a set of size a and B a set of size b, then the set A^B of all maps from A to B is a set of size a^b. And there is indeed a single map from the empty set to itself.

Polynomials/power series: Ever used sum (the bug sigma one) notation to write down a polynomial¿ Because it is very standard to write them as a sum over a_i x^i, and the term with i=0 is the constant part. But that makes only sense if one lets x^0 be 1 all the time, not only for x not 0.

c) By a more general and also very useful convention the empty product is always 1. As 0^0 is an empty product of 0s, we get it to be 1 again.

d) A lot of laws and rules simply carry over. Actually every single one I can think of that is at least possible to be satisfied (continuity is not, for example) and true for nonzero a,b in a^b.

Well, in truth, it can be anything. 0/0 follows the laws that anything over itself equals 1, so this applies to 2/2, 3/3, 4/4, ect. This means that 0/0 equals basically everything that can be put over itself and equal 1.

Now, this does conflict with what I said earlier. Nothing can be divided by 0 because it would lead to infinity and start causing paradoxes.

While this may be true, it is not the same situation as this, exactly, because, while infinity may not be equal to itself, 0 is.

0/0 equals anything that can equal 1 when put over itself. So basically, everything.

I have no idea what you want to say here. "0/0 equals anything that can equal 1 when put over itself" sounds at least random as you gave no reason for that. And "can equal" is a weird construct to begin with.

Anyway, there is not much reason to think about 0/0. Or of 1/0 as a real number. But it is not true that 0 is a "dangerous thing". The only thing that is problematic is that amateurs use functions (e.g. division) at places where they are not defined yet expect a meaningful result.

Edited January 29, 2015 by ZetaX

[Xannari Ferrows]

Sorry, but you're talking mostly gibberish.

Oh this should be good.

In the realm of Real or Complex or Integer or Natural numbers, the answer is simply that it is not a legal operation, and that's the end of it. Yes, you can add infinity to your realm of numbers to "solve" this problem and make division a globally defined continuous operation, but you inevitably run into problems with the other axioms you want numbers to obey.

And yes, twice infinity equals once infinity. Look up the Hilbert Hotel.

The problem with this is that you are treating infinity like a number. It is not a number, and you cannot treat it like one. Once infinite does not even equal itself.

The hilbert hotel, eh? I heard about this a while back. I actually was watching numberphile when Brendon told me about it. I went back to see if they have anything on this subject, and they do! They actually have a video that's basically everything I've talked about here, and makes the same points. you should go find them.

Yep. Your rule "0 to the power of anything is 0" is not an actual rule.

"0 to the power of anything is 0." -numberphile

"On the other hand, 0 to the power of anything else is 0." -HotMath.com.

Plus, I've known this since 5th grade, so there's really no denying it.

For Natural numbers (0 included) a and b, the only rules are:

1. a⁰=1 for all a

2. a^b=a*a^b-1 for b > 0 and all a.

For integer/whole numbers, you extend rule 2 to all b and you're done. So no argument there, according to the real rules, the only valid value for 0⁰ is 1 in those domains.

Not really. While the instated rules are true, you're using them in a way that ignores the fact that 0^x, where x is anything, is 0. 0^0 can be 0, 1, 6, 5 billion, or 21 quintillion, and it would all be true.

Now, for Real numbers, if you want, you can define a^b as a completely new function and depending on how you do it, it can be consistent to have 0⁰ equal either zero or one. However, if you operate on real numbers and a result of your calculation is 0⁰, that means you have at some point not done a proper limit process; and depending on what that proper limit process would be, the true answer you seek could be anything (a^b is only continuous and well defined for all b if a > 0), so the value you assign to 0⁰ is somewhat arbitrary and of little significance. Usually, one would pick 1 for consistency, then.

So, you're saying that we should stick to the true number line instead of the complex number line? I mean, everything up until there points to 1 as your answer.

Fair enough. This is true, but as a fact separate from thread.

See question 1. Not legal. Possible to define away, but you get other problems. If your calculation result is 0/0, then you again forgot a proper limit process, in which case I refer you to kerbiloid's answer.

I don't see how this works with the rest of the post? It just supports what's below.

Issues with 0 can never be defined as impossible, even though they make no sense. All we can do is speculate. (This means that everything I just said is correct and incorrect.)

- - - Updated - - -

Apparently I need to make some points clear to everyone before you post again.

This ENTIRE post is nothing but an idea. I'm not stating anything as a fact unless I'm using it as an example or stating a law. I will not state a law unless it has been proven.

I will not use something as an example if it is not relevant to the topic or if it is wrong. I read up on these things to make sure that what I'm as these things is true.

EVERYTHING I talk about that involves infinity can be resolved by imaginary numbers. I am not talking about that, so please keep it out of here.

EVERYTHING that is concluded in this post is not true. Once again, I am not stating it as fact. If there were facts about it, this post wouldn't exist. Hence: "Issues with 0."

Whatever I say CAN be contradicted, but that goes both ways as well.

This post is called "Issues with 0, and why it is dangerous." And it clearly is...

Keep that in mind.

- - - Updated - - -

Zero is not a real number, but just an idea? Any number is just an idea. And what Z-Man said. It is all defined pretty well and consistent in mathematics. We had hundreds of years to think about all this.

That makes some sense. I mean, the origins of 0 made no sense to early mathematicians. There was really no point in having a representation of nothing. In fact, there's not really a point even today if you think about it.

[ZetaX]

You did not respond to my very long previous post which contains many things not mentioned by others. Especially a lot of good reasons why 0^0=1.

"0 to the power of anything is 0." -numberphile

numberphile is often doing faulty mathematics, and that's just one more example.

Not really. While the instated rules are true, you're using them in a way that ignores the fact that 0^x, where x is anything, is 0. 0^0 can be 0, 1, 6, 5 billion, or 21 quintillion, and it would all be true.

That's not a "fact". By the way, you calling that a fact is somewhat contradicting your claim to not call such things facts, but "ideas".

Issues with 0 can never be defined as impossible, even though they make no sense. All we can do is speculate. (This means that everything I just said is correct and incorrect.)

One can define it as "impossible" as one likes. Definitions are not "speculation". And I don't see what your point is.

There was really no point in having a representation of nothing. In fact, there's not really a point even today if you think about it.

Reaaaallllllllyyyyy......................

[Xannari Ferrows]

You did not respond to my very long previous post which contains many things not mentioned by others. Especially a lot of good reasons why 0^0=1.

numberphile is often doing faulty mathematics, and that's just one more example.

That's not a "fact". By the way, you calling that a fact is somewhat contradicting your claim to not call such things facts, but "ideas".

One can define it as "impossible" as one likes. Definitions are not "speculation". And I don't see what your point is.

Reaaaallllllllyyyyy......................

There is no point in arguing with you. You will not say official and professional sources are doing faulty mathematics when you can say nothing to prove it, and I , as well as many others can back them up.

Read the post update below for some extra information. You will not label ideas as definitions. Everything you know about that is wrong.

[N_las]

The problem with this is that you are treating infinity like a number. It is not a number, and you cannot treat it like one.

You should really stop talking about thinks you have no idea about. As said earlier: Numbers are just ideas. That is the reason that there are so many number spaces. In some space, only rational numbers are "numbers", in some spaces, only real numbers are "numbers", in some places only complex numbers are "numbers". You think in a space where you exclude infinity from the group you call "numbers". But "numbers" is a concept invented by humans, and we can just as easily define a space were infinity is a number.

"0 to the power of anything is 0." -numberphile

"On the other hand, 0 to the power of anything else is 0." -HotMath.com.

Plus, I've known this since 5th grade, so there's really no denying it.

Yeah no ....... as explanation for five graders this is sufficient. But the simple example of 0^(-1) does show that it simply isn't true.

So, you're saying that we should stick to the true number line instead of the complex number line? I mean, everything up until there points to 1 as your answer.

Fair enough. This is true, but as a fact separate from thread.

You don't even know what complex numbers are. They have nothing to do with this topic.

Issues with 0 can never be defined as impossible, even though they make no sense. All we can do is speculate. (This means that everything I just said is correct and incorrect.)

Impossible is the wrong word. Use "undefined". It isn't impossible to divide by zero, the result is just undfined. But numbers are just ideas. I can easily invent a number space were 1/0 is defined as the value g. I don't think that there would be any meaningfull use for it, or if it would be easy to define consisten rules or that space, but that is besides the point.

EVERYTHING I talk about that involves infinity can be resolved by imaginary numbers. I am not talking about that, so please keep it out of here.

Imaginary numbers have nothing to do with infinities. And they have nothing to do with what your are talking about.

This post is called "Issues with 0, and why it is dangerous." And it clearly is...

Keep that in mind.

The only thing your post is demonstrating is that it is dangerous to use mathematical operations if you don't know how they are defined. The only issue with dividing by zero is that some people are treating that as a scientific questions and there were some answer we could find. It is just a question of our definition.

[rtxoff]

I remember a similiar thread like this one: http://forum.kerbalspaceprogram.com/threads/98275-What-s-the-difference-between-0-and-0

Both went out of reason and both won't lead to any new and useful conclusions.

I suggest to close this madness.

[Xannari Ferrows]

Man. When I said 0 was a dangerous number, I wasn't kidding. Brady was right as well. People argue about it, but the fact is we will never know the answers to these questions. All we can do is guess.

Well, I'm going off to talk with some civilized people who actually know things. Bye to you all.

(Plus, I really don't care if this is closed. In fact, I think it's for the best. This was originally intended to be an informal thread, but it went off the rails very fast because of the fact that people will just argue about things they simply cannot prove and do not know themselves.)

[sal_vager]

I remember a similiar thread like this one: http://forum.kerbalspaceprogram.com/threads/98275-What-s-the-difference-between-0-and-0

Both went out of reason and both won't lead to any new and useful conclusions.

I suggest to close this madness.

Man. When I said 0 was a dangerous number, I wasn't kidding. Brady was right as well. People argue about it, but the fact is we will never know the answers to these questions. All we can do is guess.

Well, I'm going off to talk with some civilized people who actually know things. Bye to you all.

(Plus, I really don't care if this is closed. In fact, I think it's for the best. This was originally intended to be an informal thread, but it went off the rails very fast because of the fact that people will just argue about things they simply cannot prove and do not know themselves.)

I agree, rather than produce anything useful this thread and the one before it is just turning to argument over method and accusations over ability.

No one is more "right" simply by shouting louder and dismissing others.

It seems that discussions of the value of zero have zero value here.