…then what makes zero unique of all real numbers?
- You can’t divide by it.
- It’s neither negative nor positive.
- It’s the additive identity.
- It’s the name of a Japanese airplane.
- without it, you couldn’t say “drop the zero and get with the hero”
It’s the only number n where m*n=n, no matter what m is.
It’s the additive identity, i.e. x+0=x, for all x.
x^0 is 1 for all x.
And, as you said, it’s the only number you can’t put on the denominator of a fraction and have it be defined.
Without him and 99 of his brothers, we’d still be searching with Infoseek.
The reason you can’t divide by zero, on a standard account of what it means to divide, is the reason TJdude825 provided. Division by x is supposed to be the inverse of multiplication by x. Thus, (yx)/x is supposed to equal y. However, since 10 = 20 = 680 = …, this is an impossible condition to nontrivially meet, at least with regards to the particular division 0/0, as we would be led to 1 = (10)/0 = 0/0 = (20)/0 = 2, etc.
Incidentally, I think people are misinterpreting the OP. It seems to me, he’s not saying “Tell me, what are some nice properties uniquely held by the number zero”; he’s saying “Why on earth, out of all the real numbers, would just one of them be incapable of being divided by? Why this bizarre asymmetry?”
Concur.
The difference, then, is the same as the difference between ‘some quantity of apples’ and ‘nothing’. Note that ‘nothing’ isn’t ‘no apples’, because any of the properties of apples are only relevant when you actually have some apples - the properties of apples are absent when the apples themselves are absent.
With all other numbers, you have a quantity, with zero, you have nothing - not a small quantity, an absence of quantity…
*Also it’s ‘nothing’ rather than ‘no apples’, because otherwise we could observer that it’s exactly the same as ‘no bananas’, and infer than apples and bananas are the same.
I’m not at all sure what your description of the difference means, Mangetout. What do you say about real numbers like -1.38 and our ability to divide by them?
To me, quite simply, the reason that we can’t divide by zero but can divide by other reals is because multiplication by other reals is injective but multiplication by zero fails to be, in the worst possible way. That is, distinct numbers produce distinct products when multiplied by nonzero values, but all produce the same product when multiplied by zero. It’s no wonder, then, that multiplication-cancelling can be undertaken for all and only multiplication by nonzero factors.
I think that’s actually similar to the idea I was trying to express.
An example of why it is so unique. I’m a machinist. One of the tools I have for measuring parts are digital calipers. A sliding scale that has a digital readout - reading to .0005 of an inch (accurate to close to that… 
)
The calipers actually have two zeros. -0.0000 and 0.0000. One for 0 to -0.0005 and one for 0 to +0.0005. But in strict mathematical fact, there is no true zero. The actual “zero measurement” has no bounds. As a point on a number line, it has no width, no size. It is a theoretical point. It is infinitely nothing. Thus it is technically real (or more correctly an integer) but it is also not “real”. “Zero” doesn’t exist. Almost into the quantum physics realm.
That’s what I was trying to get at - zero isn’t just a very small number, it’s nothing. And ‘nothing’ behaves differently from any quantity of ‘something’.
Yeah, but you could have said:
But in strict mathematical fact, there is no true twelve. The actual “twelve measurement” has no bounds. As a point on a number line, it has no width, no size. It is a theoretical point. It is infinitely nothing. Thus it is technically real (or more correctly an integer) but it is also not “real”. “Twelve” doesn’t exist.
Ditto for any other real number.
Methinks that the width of the number on a number line is irrelevant to the current discussion. I much prefer what others wrote about zero really being “not a quantity” rather than “a zero quantity”.
Zero is the only real number that doesn’t have a multiplicative inverse (a.k.a. reciprocal). For 8, there’s 1/8; for 2/3 there’s 3/2, for –20 there’s –1/20; for 1 there’s 1. But this doesn’t really answer the question so much as reformulate it.
I think the answer (insofar as there is one) is this:
Since 0*anything = 0, you can’t “undo” the multiplication.
This is a consequence of 0 being the additive identity (that is a+0=a for any a), since:
a+0=a
then for any number b: (a+0)b=ab
so ab+0b =ab
then obviously 0b=0 for any b.
So I’d say its the fact that its the additive identity that ultimately causes it to be invalid as a denominator. Which makes sense, since I believe that 0 is usually defined as the additive identity in most math courses.
Unbreakable encryption!
Ah, then so maybe the OP would be interested in the concept of a field in algebra, and with the fact that in any field (not just the reals), every element has a multiplicative inverse, except the additive identity (usually called zero). The reason why the zero doesn’t have a multiplicative inverse is because for every element x in our field, 0 * x = 0, as mentioned by Indistinguishable. This property is easy to prove using the other properties of addition and multiplication in a field:
0 * x = (0 + 0) * x = 0 * x + 0 * x, and subtracting 0 * x on both sides we get 0 * x = 0.
To learn about fields, I guess the Wikipedia article is as good as any other place to start.
Sounds like a no-brainer to me: You can’t divide by zero, therefore zero is unique because you can divide by any other real number.
I think maybe we should put further discussion and speculation on hold, until such time as Lumpy tells us what his question really means.
You’ve all got it wrong. It’s the number one that’s unique. Zero is zeroque.
If you find this thread interesting, both for its mathematical and wider philosophical implications, you may enjoy Zero: The Biography of a Dangerous Idea.