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The Concept of Zero

  1. May 5, 2012 #1
    Hello, I'm sorry if I'm posting in the wrong place. As you can see this is my first post and there's something that's been irking me so much that I need clarification. I'm sure some people will be disgusted at some of the logic, but I just want clarification. I do realize that it's all hypothesis with no real mathematical proof, I simply don't have enough knowledge in mathematics to even get close to trying. I'm all ears though, so if there is some mathematical formula to shed some light then I'll try my best to understand it.

    The basic question I want answered is, "Is Zero a real number?" (insert gag reflex)

    My train of thought first came from why you can't divide by zero. You can divide something by a number that infinitely approaches zero and it approaches infinity when you look at any graph. As in, x/ε = ±∞. The same can be said for the inverse, so x/∞=ε.

    I remember the first thing I learned in Calculus was that any number that infinitely approaches a number can be considered the same, as in .999999999......= 1. So why doesn't it work with zero? I just hear people say, "well, zero has a few exceptions because it's an identity element". As far as I can recall, 1 is also an identity element and it's keeping itself in line just fine. So then I thought, "Is Zero a real number?"

    There's actually another "number" in mathematics where it infinitely approaches the number yet it can't be considered the same, and that is in fact "". Why is something infinitely large not considered a real number, yet something infinitely small is considered a real number?

    So then I thought, "Maybe zero is a concept and it's not a value you can actually reach, just like ∞". This makes me think that there's a good possibility that zero shares more traits with infinity than a real number. This means that 5+0, 5-0, and 5*0 are all something you simply can't do, just like infinity. If you look at it another way, adding, subtracting, and multiplying "nothing(zero)" might also be impossible since it isn't actually a value. By this logic, every instance of zero in current mathematics was actually always ε, so 5+ε=5, 5-ε=5, and 5*ε=ε. I pondered the thought, and found myself deducing that we wouldn't notice any difference whatsoever. I did some brief research on infinitesimals, and I'm pretty sure this is just a different way of approaching non-standard analysis. It was really brief research though, and I don't have any idea where to start.

    So, why is it important if it doesn't make a difference? To be honest, I don't know. This was the same treatment that non-standard analysis got, in that it was just a different way of achieving the same answer. However, it might make a difference somewhere that I'm not aware of though, like quantum mechanics or something. Maybe non-standard analysis can get somewhere that normal calculus can't because of that little difference.

    Of course, I could be completely and utterly wrong somewhere. Criticism much appreciated.

    P.S. I'm not trying to prove anything with this post, I'm looking for something that can clarify it a little more. I realize that I'm just assuming zero is more similar to ∞ than a real number without any proof whatsoever.
    Last edited by a moderator: May 5, 2012
  2. jcsd
  3. May 5, 2012 #2


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    Hey HHayashi and welcome to the forums.

    One thing that will help you in terms of what a 'number' is, is that for most systems numbers are measureable. This is one reason why in many cases, infinity is not considered a number.

    In terms of your arguments like 5-0, 5+0 and so on, again it's probably better to take a page out of the geometry book.

    Instead of thinking about numbers as symbols, think about numbers having a length. Then think about what happens when you add those 'lengths' when you associate a direction with those lengths. This is what vectors are all about: vectors have length and a direction.

    Negative numbers point one way for a one-dimensional vector and positive point another way. A 0 vector has no length and no direction. When you add these arrows together, you also add lengths like you would expect and because 0 has no length, it won't change the result of a+0 where a is a vector.

    Now remembering things in terms of arrows and the associated lengths and directions will help you make sense of zero. It will also help you make sense of a-a or -a + a as well.

    In terms of division, it turns out we can actually define what division involving 'arrows' actually means as well but this is a lot more complicated. This is actually the basis for a lot of modern mathematics and it also relates to what are called complex numbers which are really important in every area of science and engineering.
    You should remember that mathematics in many respects grew out of the practice to measure stuff and the word geometry literally translate to 'earth measure' or 'the practice of measuring the earth'.

    Numbers essentially capture variation. The ability to do this is what mathematics is used for. Initially we captured variation only in numbers, but then we soon applied the idea to functions and all kinds of things and we are adding different kinds of generalizations to higher abstractions and concept within mathematics.

    Because numbers are measurable, no matter how big or small, we discard infinity from it being a number. Infinity is more like a concept than a number and it is very useful in mathematics as being a concept because it helps us understand what the real nature of infinity is and if you look at modern mathematics, you will see that understanding infinity is not an easy task.

    Finally don't confuse a limit with an actual fixed number: they are not the same.

    Limits are a great idea that says that we will consider what happens when something gets extremely close to something, but it doesn't actually become something. In other words they are not the same, but they get closer and closer without actually becoming the same.

    The above idea is what a lot of modern mathematics is based on and it is what calculus as a whole is based on. It also turns out that with this kind of thing you can define whether something is continuous. differentiable and so on in a general way and this is pretty powerful mathematically.

    So yeah think about things as arrows with length and direction and also think about limits and the fact that infinity is not a number because in the practical sense, it can't be measured and numbers have to be measureable.
  4. May 5, 2012 #3


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    Some of your questions are answered in the FAQ: https://www.physicsforums.com/forumdisplay.php?f=207 [Broken]
    Last edited by a moderator: May 6, 2017
  5. May 5, 2012 #4
    Thank you both. Thinking about continuity reminded me about all those times in school where I solved a function, and then added x≠0 at the end. I'm kinda glad that some of this stuff is still at least partially coming back to me even after 7 years of not touching it. :smile:

    I am extremely intrigued by this part, and would like to know in more detail. This would actually be very strong proof that zero is not an attainable value, since you can't ever actually make 0.999999999.....=1, just infinitely close. I read the FAQ on this part of the subject, and this seems to be the case in proof #2 at least, as it uses infinite geometric sums to prove it when infinity is also an unattainable value.

    This part, on the other hand, kinda messes things up. If my logic is correct, then a-a=ε. This means I can never fully take 5 apples out of 5 apples in any context, whether it's spatial or making a pie out of it. That sounds intuitively messed up, unless we can embrace the viewpoint that there's a little bit of apple in all of us and it can never be fully taken away. ...Actually, that doesn't sound all that impossible since we're all made up of atoms. It's still REALLY far-fetched though. I'll try to make some sense about it, although a-a=0 does seem intuitively correct.

    As for the rest of it, my logic is simply suggesting that zero may not be an attainable value. Thinking about it in different contexts such as length doesn't really change anything since I'll just apply that zero is not an attainable length. In your example, you say that numbers are measurable in most systems. I dunno about you, but I cannot possibly measure 0 centimeters.

    In other words, you can't ever completely have "nothing" of something, whether it's length, velocity, time, etc. Well, maybe you can once you start getting into derivatives. I'll need to contemplate that part. That, and a-a.

    To clarify something, I'm very aware that infinity is not a number. It makes a whole lot of sense to me that it isn't. I'm here to share my idea that maybe zero is not a number either. We've lived our entire lives being told that 5+0=5, 5-0=5, etc. because it all makes sense on the surface. However, when I look deeper I'm not so sure. With the discovery of atoms and quarks, it doesn't seem completely impossible if I was told that I can never completely take 5 apples out of 5 apples.
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  6. May 5, 2012 #5


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    No, we DO have 1=0.99999...
    And if two things are infinitely close in the real numbers, then they are equal.

    a-a=0 by definition. You can't reason your way out of a-a=0, it is simply true because we want it to be true. If you want to change the definition of a-a=0, then you may do so, but no mathematician will follow you in this.

    The only question you can ask yourself is: is a-a=0 a useful definition?? It appears that it is as it is used in physics and science every single day without problems.

    We work with 0 not because it is physically possible, but because it is a useful definition. You can do math without 0 if you want, people have been doing just that for thousands of years. But it appears that 0 makes everything a lot easier.

    Define number.

    This is physics, not mathematics. In mathematics, we define 5+0=5 and 5-0=5 because it is useful to do so. Physics tells us that such a definitions are useful.
  7. May 5, 2012 #6


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    As has been said in this thread and the kinds of threads micromass posted, infinity is something that is not intuitive for many people that have not studied it because it is a concept and not something that is easily physically tangible. To understand it, you have to resort to using mathematics and similar things and when you study it, you find that it really is something that comes with a lot of 'buyer beware' statements.

    The number 0 is a finite number. When you deal with arithmetic involving finite numbers, you don't need to resort to using limit definitions (unless you are doing something like a division by zero or something similar to this).

    Again when we are dealing with finite numbers with +,-,* operations, think of the numbers as arrows that have length and direction and it will make a lot more sense. Don't just think of them as quantities that have no direction, because this will make it more confusing.

    There is a physical intuition behind 0, but this is not the only intuition. If you think about arrows, then what you do is interpret what the arrow means. If you have 0, you don't have any arrow because it has no length and the direction itself is unresolvable because there is no length: instead what you get is a point. As soon as you have length, then you need to also specify a direction since you can only have 'one arrow'.

    Because of the fact that zero has no length and no direction, you can't really say it's an arrow, because if it was technically an arrow it could point in any direction it wants to since it has zero length.

    You can use the above kind of analogy to see why you get a lot of problems in mathematics with 0 in other contexts (including division by zero) and also with other areas of advanced mathematics like linear algebra and higher abstract algebras.

    With regards to the pie, you can take 5 apples away from apples by representing your original five applies with an arrow with length five in one direction and then -5 apples by an arrow with equal length but in the total opposite direction. When you add the two arrows together you end up getting the 0 vector which is just a point with no direction or length.

    You can interpret this zero point in many ways which include 'the absence of anything' or 'the equilibrium' or something similar. Remember that if we have 'something' then we have an arrow with length and a direction, but if we have 'nothing' then we have no arrow with a direction.

    This might help you understand what 0 actually means.

    Hopefully the above has helped you with this question as well.
  8. May 5, 2012 #7
    Thank you both for your replies. There seems to be a little gap in between what you guys think I'm trying to say and what I'm actually trying to say. Or maybe I just don't understand what you're trying to say, and I'm sorry if I don't.

    I guess I'll define number as what you describe, "something of measurable value". Zero certainly doesn't seem to fit in that category, although ε isn't either.

    I'm not trying to disprove a-a=0, I'm trying to say that our understanding of zero may not be accurate (or maybe I just didn't get far enough in mathematics). In other words, nothing in mathematics will change at all if a-a=ε, as long as we replace every zero anywhere with ε and don't mix the two together.

    In other words, you can do everything in math with ε that you can with 0 and vice versa. I don't deny that the usage of zero is very convenient, and I don't know what difference it makes if 0 was replaced with ε.

    In your case of vectors, all of it still makes complete sense to me if 0 was replaced with ε for everything you said. I'm fine with using 0 as the placeholder for 'equilibrium' and 'absence of anything'.

    It seems asking if zero was a real number was the wrong question. I'll change my question: "Is zero an attainable value?"
  9. May 5, 2012 #8


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    OK, under that definition I think it makes sense not to make 0 a number. However, you should be aware that this is not the definition that mathematicians agree to. That something should be measurable is a notion that was popular in ancient Greek mathematics, but it seems superfluous now.

    It's ok to think of "0" as something which can not be measured and something which might not exist in reality (this is arguable though-. But it's still easy and correct to work with it. It will still give us the right answers. So perhaps you should think of 0 as a shorthand of other things.
    It's a bit like with imaginary numbers. These might not exist in reality, but it's still nice to work with.

    It is not clear to me what ε is supposed to be. How did you define it??
  10. May 5, 2012 #9
    I was using ε as the concept of something that is infinitely small, from the equation 1/∞ = ε.

    I do agree that whether it's accurate or not, using 0 is still easy and correct for all purposes I can think of. That's why I dunno what difference it makes.
    This is the part I want to know. I want to hear that argument.
  11. May 5, 2012 #10
    You can't disprove it because, maybe it was brought forward as a definition, but it is true both if 0 is intended as number, as nothing and as empty space; both if you consider - as a reverse operation and if you consider -a as a negative number: a + [-a] = 0. Positive and negative numbers annihilate themselves. There is nothing, 0, zero, nil, nought, the empty page/space:
    [ ] , then we have
    1, we add (-1) 1 ,
    1 1
    [ ]
    If you consider - as the reverse operation of + (giving, writing, creating)
    [ ]
    you write
    1 , then rub it out
    [ ], 0
    zero, nil, nothing the empty page
    micromass, same question I asked for
    1) if we decided not to make 0 a number under any definition would there be any disadvantages?
    2) it is not arguable, that's one of the few sure things in the world, 0 (unlike ∞) has no ontological status, it does not exist in principle and it cannot exist anywhere, anyhow because 0 is 'that which is not' is a contradictio in terminis, it's the place .. where all the cats are grey :smile: (2*0 = 3*0; 2:0 = 3:0...)
    Last edited: May 5, 2012
  12. May 5, 2012 #11
    I see. That shed a little more light on what I'm actually trying to say.

    Let me start out by saying that my stance on mathematics is the same as what I see in wikipedia: the study of quantity, structure, space, and change. As such, I believe anything that happens in the actual world must be applied in mathematics. However, since math is conceptual the same cannot be said the other way around.

    I am also a firm believer in that two things cannot possibly be completely identical to each other. If you look infinitely deep enough, there is bound to be something different.

    By extension, this means that although a and (-a) annihilates each other, there cannot possibly be anything that is completely equal or completely opposite of something else. Which implies that 1-1 or 1+(-1) cannot actually ever equal zero.
  13. May 5, 2012 #12


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    That's more philosophy than actual mathematics.
  14. May 5, 2012 #13
    It is not philosophy, only a fallacy, a little misjudgement:
    (yor premise 'no 2 things are identical' might be true, but your) conclusion is not because : I give Hayashy and take away that same, identical Hayashy, I get the status quo ante, perfectly identical. Dad gives you a PC and , if you are naughty, takes back the present. 1(PC) -1(PC) = 0 (PC)

    {a + [-a]} , a math invention, ruse, figment; annihilation, a figure of speech: 'the anti-number', just to explain why it is possible 'in principle', in abstract
    Last edited: May 5, 2012
  15. May 5, 2012 #14

    It'd be interesting to know whether you can give a definition that makes sense, let alone mathematicswise, of what "looking infinitely deep enough" means...

  16. May 5, 2012 #15
    Well, this is seriously getting more into philosophy but here goes.

    Time, as defined, is a continuous progress of existence. It never stops. Assuming time travel is impossible, there can only be one of me at any given moment so you cannot add or subract me from me. Even if a fraction of a second changes, it's a whole new me and there will be a difference.

    In mathematics, "looking infinitely deep enough" is basically finding the difference between .99999.....and 1. "Looking infinitely deep enough" is what infinite geometric sums tries to do.

    Now that I think about it, my idea is trying to prove itself. If zero is not an attainable number, then there is certainly a difference between .9999999... and 1. If it is an attainable number, then there is no difference. Hrrrrmm....
  17. May 5, 2012 #16


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    There is no difference, and this has been the subject of many discussions in this forum. Here are a couple of links:
    https://www.physicsforums.com/showthread.php?t=507002 [Broken]
    https://www.physicsforums.com/showthread.php?t=507001 [Broken]

    I will agree that there is a difference between .9 and 1, and between .99 and 1, and even between .999...9 and 1. However, as soon as you add the ellipsis (...), then .999... and 1 are equal.
    Your implication is not valid. When you have a conclusion that is false (there is a difference between .9999.. and 1), it doesn't matter whether the hypothesis is true or false -- the overall implication is invalid.
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  18. May 5, 2012 #17
    I was questioning the possibility for there to be zero "distance" between the two numbers. "0.999...=1" proof is under the assumption that "0.999..."can go infinite. My hypothesis is trying to prove itself because there's no difference between ".999...." and 1 if it can. In mathematics, I'm sure it can. Can it go infinite under any applications in the actual world though?

    Edit: Actually, maybe it isn't trying to prove itself. I'm starting to feel that unless you can reach infinity in any, way, shape or form (which mathematics says you can't), you can never reach zero either.
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  19. May 5, 2012 #18
    An expression such as .999... is a shorthand for the infinite series

    9/10 + 9/100 + 9/1000 + ...

    It "goes infinite" by virtue of the definition of an infinite series; which is based on the definition of an infinite sequence; which is based on the definition of a function.

    It's mathematics.

    When you ask, "can something go infinite in the actual world," that's no longer math. I'd say no. There are no infinite sets in the physical world.

    Infinite sets are something we work with in the abstract, conceptual realm of mathematics and in particular, set theory.

    That should not trouble you too much though. As human beings we are very good at working with abstract conceptual things. Fictional novels, video games, movies. Concepts like justice and love. Those are abstract conceptual things yet they are supremely important to people.

    Set theory and math are the same way. They're a video game or your favorite cop show on tv. You may love your favorite tv character and be able to talk about them and argue about whether this behavior or that is consistent with their personality. But you never bother to say, "Oh I can't really talk about this, it's only a story."

    Set theory is a story. Whether it applies to the real world? That's philosophy. So far there's very little if any evidence that there are any infinite sets. But that doesn't mean they're not useful; just as the abstract concept of justice is highly useful in determining who we should lock up or execute. Justice is an abstract idea that's very difficult to pin down in words; but it is the basis of life or death decisions in the real world.

    Perhaps we can view infinite sets the same way. Fictional, but extremely important because they provide a framework for doing math, hence physics, hence flying to the moon.

    So even though there are no infinite sets in the real world; they're very handy when doing calculus and plotting the course of space ships and missiles.

    How can something fictional be so useful? There's your topic for your philosophy thesis. It's a mystery. Like justice, or love.
  20. May 5, 2012 #19
    How many 10ft tall blue elephants live in your mouth right now?


    QED ; )
  21. May 5, 2012 #20


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    What do you mean "my hypothesis is trying to prove itself"? A hypothesis is simply a mathematical statement of some kind. It can't "prove" itself. It is either true or false.
    The function f(x) = 1/x approaches zero as x gets large. Mathematically, this is stated in terms of a limit:
    $$\lim_{x \to \infty}\frac{1}{x} = 0$$

    For any large, finite value of x, 1/x will be small; the larger x is, the closer 1/x is to zero. You can get ever closer by choosing larger values of x, but as long as you pick a finite number (and you cannot simply choose ∞), the value of 1/x will be different from zero. This might be what you're saying in the last paragraph above.
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