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What if 0 is just a concept with no actual real-world counterpart?

  1. Jan 3, 2010 #1
    Have you ever noticed that when you get 0 involved in even the simplest multiplicative operations, it manages to screw them up? For instance, take the functions f(x)=1/x and g(x)=xx. As for f(x)=1/x, f(0)=1/0, an expression that, from one side of the equation's graph, appears to represent positive infinity, but from the other side appears to represent negative infinity. And what about g(x)=xx? That seems to be a pretty straightforward operation. Yet, g(0)=00; conventionally x0=x/x, but 0/0's value is context-dependent, and no context was set by use of g(0); hence, g(0) could mean anything depending on its own context, but by using it in two different contexts, we could get, say, g(0)=2 and g(0)=3, and since 2[tex]\neq[/tex]3, one could conclude by substitution that 0/0[tex]\neq[/tex]0/0. But what about just using 2 or just using 3? 2=2 and 3=3, so substituting from either of these cases, we could find that 0/0=0/0. So there you have it: an expression that looks like two completely different terms depending on what side you approach it from and an expression that both does and does not equal itself. Weird, right? Why is it that all the operations in math that are illegal, that have no answer at all, all seem to trace back somehow to li'l ol' harmless 0? If so many things related to 0 are illegal operations, might it then be possible that in mathematics that run completely parallel to the functioning of the universe, 0 itself should not exist? Back when 0 was first invented, I'm certain it was during the early years of mathematics, where it had much more to do with the real world, back when it was used for tallying; 0 was probably used to represent, for example, the complete absence of oxen to pull carts, or chickens to lay eggs, or maybe just bananas. But is there ever a complete absence of bananas? Isn't it possible that no matter how many bananas you take away, there's always just a little tiny bit of banana left over? You can't really get rid of bananas completely; they're dropping atoms and molecules all over the place with each step you take while carrying them. Parts of themselves. Fractions of bananas. Maybe even if you try to take away parts of those atoms and molecules, there will always be little bits of them left. Maybe everything trails infinitesimal pieces of itself wherever it goes. Why does there have to be a 0, a point where is absolutely nothing? What if there's never absolutely none of something? If that's the case, 0 shouldn't even exist, and if it does, it's a mathematical concept with no real-world parallel. Of course, all of this is hypothetical; if I knew any of this for sure, I wouldn't even be talking about it here.

    You have 6 dollars. You want to split the money amongst no friends. How much does each friend get? Well, obviously, 6/0 dollars. But let's just talk about division by things less than 1 in general: You have 6 dollars. You want to split the money amongst half a friend. Yeah, I know, it's gross, but let's just say you're a creepy serial killer guy. How much does the half-friend get? 6/0.5 is 12, but the half-friend can't possibly get 12 dollars; you don't even have that much. Analyzing things this way, I see that dividing money up amongst friends when you have less than 1 friend isn't really what it seems; sure, it's giving an equal amount of money to each friend, but when we have less than one friend, we must consider the definition of "each friend." If you have a half a friend and you're giving the half-friend some amount of money, then if you had 1 friend, that friend would be two half-friends, so you'd be giving that friend twice the amount of money as you'd give a half-friend (to be fair to the other two half-friends that make up the 1 friend). If you only have 6 dollars, of course you're only going to give the half-friend 6 dollars; that's all you've got, and the half-friend is your only half-friend that's there. That means that if you had 1 friend, you'd give that 1 friend 12 dollars. But even though you don't have 12 dollars, you don't have 1 friend. So you don't have to worry. So now that I've demonstrated that if you have a half-friend and 6 dollars, he doesn't get 12 dollars, he gets 6, let's say you have no friends. You're a sad little fellow, aren't you? Each of your 0 friends gets 6 dollars. There you go, 0 friends. They get 6 dollars. You lay the 6 dollars down at the location of your 0 friends. Nothing happens. No-one takes it. As you stare into the money, it seems to stare back, asking you to take it back. But you don't. Now it belongs to your 0 friends. If you ever have 1 friend, it's only right to give every instance of 0 friends in your 1 friend 6 dollars (to be fair). How many 0 friends are in your 1 friend? 1/0. Multiply that value by 6 and find 6/0. This means that if every group of 0 friends in your 1 friend gets 6 dollars, your friend should get 6/0 dollars. The problem is, no matter how many groups of 0 friends you have, you still don't have a single friend. You poor soul. You must be so depressed constantly putting together groups of 0 friends. You know, Einstein once said that insanity is defined as doing the same thing over and over again and expecting a different result. Like you're doing. You poor, insane soul.

    But wait!

    Have you ever been near people? Hey, at least you've been near yourself. To not be near yourself is impossible. Everywhere you go, you drop cells, little tiny portions of friends. And even if nothing drops cells, things might drop atoms. And even if nothing drops an atom, who says things can't drop electrons? Or quarks? Or really, really, REALLY freaking small things that you can't see? So maybe you're never alone. Perhaps there's a faint glimmer of hope for you after all.

    See what I mean? Maybe we're never alone. Maybe there's never no bananas to eat, no chickens to lay eggs, no oxen to pull the cart, no friends to give out money to. Maybe 0 is nothing but a concept with no real-world truth behind it.

    ... Of course, that means that you can never take everything away from something. And if you can never take everything away from something, that must mean that subtracting a number from itself is also nothing but a concept with no real-world truth behind it. That's... kind of lame if you ask me.

    Come to think of it, I guess even if 0 is just a concept with no actual real-world counterpart, I bet nobody will care. Even if there can't ever be none of anything, pretending that there can seems to be working well enough for us all.
    Last edited: Jan 3, 2010
  2. jcsd
  3. Jan 3, 2010 #2
    Anything divided by 0 is not context-dependent. You're thinking about limits, which is wrong. That's answering the question of approaching 0 from a specific side. Plugging in 0 is much different.
    What you're trying to understand is how we define things and what makes a specific subject. Try looking up Theseus' ship.
  4. Jan 4, 2010 #3
    The same could be said for any value with infinite significant digits, ie whole numbers.
    Maybe not 1, but that's a different story.

    Just like I am never really "alone" because my friend's quarks are still hanging out with me long after my friend has left, I am never with only one friend or two friends. There may be more friend-particles hanging around.
  5. Jan 4, 2010 #4
    All numbers are concepts. Students usually get over their initial anxiety with complex numbers when they realise this. Yes the complex numbers are "made up" but so are the reals and the integers and the rationals... This doesn't make them any less useful and powerful.
  6. Jan 4, 2010 #5


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    All mathematical ideas are "concepts". And none of them have "actual real world counterparts". Since all "real world" objects are subject to measurement, which is never exact, any "real world counterpart to a mathematical concept is, at best, approximate.
  7. Jan 5, 2010 #6
    Wow. Yeah, maybe you're right. Maybe every quantity that exists is some number of infinitesimals between two irrational quantities; maybe nothing is actually some exact number of anythings except exactly what it is, of which it is obviously 1 due to the property of all non-fallacious numerical expressions z that 1z = z (assuming of course that measurements are not fallacious, which, come to think of it, I cannot prove), addressing dhagg's point of "Maybe not 1, but that's a different story." If that's the case, math can't precisely apply to the real world no matter how much we want it to, unless we're just dealing with 1's all over the place, and if that's the case, what are we doing trying to quantify things at all? I mean, sure, it seems to work to a point, but what happens when we go past that point? If all math is abstract, I guess it really doesn't tell us anything about the what the real world is going to do; I mean, sure, it can predict the past, and we can check the past against the math and it seems to add up, but how do we know that the same stuff will work in the future? Heck, if all measurements are inexact, the same stuff doesn't even work in the past; it just looks kind of like it would work if measurements were exact.

    What in the name of my aunt's cousin's brother's pet's wife's right forearm is this "limit" of which you speak? Never mind; I guess I'll learn what it means when I get to calculus. Yeah, I know, I fail, burn me at the stake, poke me with a pitchfork or whatever. DX

    But I see what you mean; I still don't get the limits thing, but actually plugging in 0 may result in the statement evaluating to 0=1:

    1/0: The number that, when multiplied by 0, equals 1 (definition of division)

    Since we all know that 0[tex]\neq[/tex]1, 1/0=1/0 must not be true, violating a property of all numbers and thus denying the status of 1/0 as a number. So yeah, I guess actually plugging in 0 wouldn't make much sense...

    Ah, here's another perspective: Multiplying both sides of an equation by 0 changes the equation's truth value, usually to that of a tautology (x=2 isn't necessarily true, but multiply both sides by 0 and you get 0=0, which is true), so since I required multiplication by 0 on both sides of the equation, is it possible that I changed the truth value of the equation by doing so, in this case to a contradiction? If so, perhaps 1/0 isn't a fallacy after all. Not that I can see any potential way to actually describe its value, but... still.
    Last edited: Jan 5, 2010
  8. Jan 5, 2010 #7
    So what you're saying is, if you can change the truth value of an equation to 'TRUE' by multiplying both sides by 0, then is it possible to change a TRUE statement to a FALSE statement by applying a function to both sides?

    it's irrelevant. What's important is that even if
    1/0 = 1/0

    is a true statement,

    the fact that we arrived at 0=1, means that introducing 1/0 in the form you propose is a bad idea, because the resulting system would be useless, because in your system 0 is equal to 1.

    Most engineers/scientists/doctors would disagree. Maths is very good at approximating the real world.
  9. Jan 5, 2010 #8


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    The amount per friend is 6/n, where "n" is the number of friends. So yes that's $12 per friend when n=1/2, so your 1/2 a friend gets 12 * 1/2 = $6, (amount per friend times number of friends).

    So ummm, what's the problem?
    Last edited: Jan 5, 2010
  10. Jan 5, 2010 #9
    Actually, that was the problem. If you read further, then you'll see that I introduced that problem in order to propose the very solution you just described. XD

    But what I'm saying is that in my system, 0 doesn't necessarily equal 1, because that is a statement that was derived from my system by multiplying both sides by 0, thus potentially changing the system's truth value; therefore, it may very well be the case that 1/0=1/0 never was a false statement (it just turned into one when we multiplied both sides by 0). On the other hand, even if the system is true, I can see your point in saying that it's useless; all it tells us is that 1/0 equals 1/0; it does not denounce any other disproof of 1/0 being a number, nor does it assign a value to 1/0 other than itself; in fact, once a value is assigned to 1/0, it makes it impossible to determine whether that value works, because you have to see if multiplying that value by 0 makes it 1, which means multiplying both sides of the equation by 0, which could mean changing the truth value. On the other hand, since in any non-division-by-0 situation, multiplying both sides of an equation by 0 seems to result in a true statement (0=0), it's possible that this is the only possible result of doing so, which in turn would imply the status of division by 0 to be a mathematically invalid operation.

    Another way of looking at why one can't find a result for division by 0 would be that by dividing both sides of an equation by 0, you're trying to perform the operation that, when performed on 0=0, would result in getting back the original equation that you had before you changed the truth value, but within the truth value of the new equation, you wouldn't have enough information to get back that original equation; as soon as you multiply both sides of an equation by 0, all the information goes down the drain.
  11. Jan 5, 2010 #10


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    Why don't you just define the system for us? That way we can comment on it directly, rather than making statements like "0 doesn't necessarily equal 1".
  12. Jan 5, 2010 #11
    (Are you suggesting that zero has zero real world counterpart?)
  13. Jan 5, 2010 #12
    wooooh PARADOX

  14. Jan 6, 2010 #13


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    ??? Where do you see a paradox? I see none at all.
  15. Jan 6, 2010 #14


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    I guess the point was either zero has zero real-world counterparts or it has more than zero. But if the first case is true, then this is an example of real-world counterpart of zero, so zero cannot have no real-world counterparts. Not really a paradox, but...
  16. Jan 7, 2010 #15
    It IS a paradox.

    Assume that there really are no real-world counterparts to zero: saying that zero has ZERO real-world counterparts would then be false, because there are 0 counterparts.

    It only works one way, though, because it doesn't work to assume that there ARE real-world counterparts. It's cool in that one way, though...

    (but still a paradox)
  17. Jan 7, 2010 #16


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    I wouldn't call a contradiction a paradox.
  18. Jan 7, 2010 #17
    I stand corrected.

    I maintain that it was still cool.
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