The Importance of Zero: Uncovering its Significance

[matt grime]

Yes, 0 is rational and it is an integer. The fact you don'ty agree is down to you not using the same definition as other people. ANd they are just definitions.

You seem to want to think numbers are some how real, as in 2 sheep. These give rise to the natural numbers, 1,2,3,... and so on. Now these are all well and good but are not closed under ALL arithmetic operations, are they? They do not even have subtraction or division. Now, we can add 0, and the negative whole numbers in. This means we have a good numerical method for saying how many sheep we have left after we've sold them all, and can even express the notion that someone owes us a sheep, or that we've oversold a sheep.

But we still can't divide in these, and can't express the relative amounts of things (1/2), so we can add in the rest of rationals giving a system where we can divide by all numbers but not zero. No, you, and many other people who think in terms of "actuial objects" go getting all funny with dividing by zero and infinity.


The oddest thing is that you think pi is more meaningful. Go and find me something that pi describes in the real world.

 

[strid]

did i say that pi is more meaningful? (although it is )

to make this more clear... can someone please tell me the definition of both integer and rational number.. ?

 

[matt grime]

So, now you're admitting you know not of what you speak?

Naturals are {1,2,3...} which we'll call N

For some people
Naturals are {0,1,2,...}, which we'll call N_0

Integers: {...-2,-1,0,1,2,...} labelled Z

Rationals: { a/b : a in Z b in N} labelled Q, with the relation that a/b=c/d iff ad=bc.

If you want to be consistenet (which you're not being), then what happens when I subtract 2 from 2? Both of those are definitely numbers, in your sense, and subtraction is an arithmetic operation isn't it?

 

[matt grime]

If you don't like that then the integers are the smallest ring without torsion, and the rationals are the smallest field of char zero.

 

[strid]

Not again.. I've said it before... I KNOW we can add and substarct with zero! I'm saying that you can't divide...

Also, I knwo what the rationals integers and all that includes ut I'm askign for a defitnion of rational numbers...


However... are you 100% positive about that

Naturals are {1,2,3...} which we'll call N

For some people
Naturals are {0,1,2,...}, which we'll call N_0

Integers: {...-2,-1,0,1,2,...} labelled Z

Rationals: { a/b : a in Z b in N} labelled Q



{ a/b : a in Z b in N} this stuff seems to be written just to include 0 into it... will check on the net myself...

 

[arildno]

strid: You cling to your own personal fantasies and just can't accept they haven't anything to do with math.
Shame on you for being unwilling to learn.

And no: You do NOT know what integers/fractions/real numbers are.

 

[matt grime]

eah, cos I'm really trying to lie to you...

the rationals are a field by construction and have a zero element.

so you can't divide by zero, why on Earth is that a problem, you've never actually articulated why that is a mathematical one, merely indicated that you don't LIKE other people's definitions. Tough. They're just definitions. IF you want to talk about the philosophiocal nature of it then go to a philosophy forum.

Reminds of that crank who disliked zero so much he "redefined" the entire positional notation of decimal representations of N so that there were no zeroes. Instead he used the syumbol A as it was "more natural" so that instead of a positional system in which ten read as 10 it was A. Complete tosh it was too.

 

[strid]

ok... sure.. you can live in ignorance and don't care of the somewhat interesting fact that 0 doenst vehave as other integers/rational numbers... ALL the arithmatic operations are applicable on EVERY rational number, but not 0... and how many times don't you exclude 0 from things just because it won't work...

I'll stop this discussion then, because people seem to not be interested in more than what their mathbook say to them, and seem to not be able to do some thinking of their own. No, because if you need to think you refuse by saying "that is philosophy". I bet none of you will never come up with something new in the mathematics...

 

[robert Ihnot]

Everyone seems to have overlooked that a great use of the zero is as a place holder when we write numbers. This was not known to the Romans and so addition was very difficult such as adding 40+21+4 = XC + XXI + IV. The zero was a great improvement for commerce.

 

[arildno]

ok... sure.. you can live in ignorance and don't care of the somewhat interesting fact that 0 doenst vehave as other integers/rational numbers... ALL the arithmatic operations are applicable on EVERY rational number, but not 0... and how many times don't you exclude 0 from things just because it won't work...

I'll stop this discussion then, because people seem to not be interested in more than what their mathbook say to them, and seem to not be able to do some thinking of their own. No, because if you need to think you refuse by saying "that is philosophy". I bet none of you will never come up with something new in the mathematics...

The fact isn't interesting at all. It is merely trivial.
Your problem is that when YOU try to expand your mind ("thinking" on your own..), it merely becomes inflated by your personal fantasies (barring a few trivialities trickling in once in a while).

Instead, you might try to actually learn something. You won't find the fuzzy, familiar warmth in that as you are used to feel in day-dreaming, but on the whole, it is by far more rewarding.

 

[Telos]

The teleological importance of mathematical notions

I assume you're not studying Mathematics currently at an academic level. It would therefore be very difficult to try and explain how important any notion in Mathematics is.

No it isn't. Abandon this notion.

State the importance of something in mathematics by immediately relating it to an applicable context, or relate its importance to another mathematical notion and relate that subsequent notion to an applicable context. If you cannot do either, your mathematics is not important (yet).

It may be simply stated, "With [mathematical notion], I can do [something]."

With the number 0, I can record, tabulate, and analyze the supply of apples in my grocery store, especially when that supply is depleted or when demand for apples is nonexistent.

With tensor analysis, I can plot a course for ship on a windy sea, etc.

These statements may sound mundane, and give little detail of the mathematical processes involved, but answering the question of importance requires, above all, some kind of telos for the thing in question.

 

[matt grime]

No, we all know that we cannot divide by zero in the real numbers, or any other field, that it is different: that is a basic exercise in field theory and one of the things we have chosen to so accept in our definitoins of fields. So? We cannot take square roots of all numbers in the reals, cannot subtract all positive numbers from each other and remain with the positives. We also all know about the extensions by continuity and some of us know about compactification to allow symbols that you'd probably want to call infinity. We can divide by zero on the riemann sphere and get infinity, and it's all well understood.

Strid, where do you get off, someone who doesn't even know that 0 is an element of Q, telling maths phds that they "won't produce anything new in maths"? Jeez, this is turning nasty. Anyone got a thread lokc available?

 

[infidel]

Hey, I'm not a mathematician by any means, but I have a book all about it which of course I cannot find now. A lot of it has to do with calculation rather than the concept of a zero.

How would you do this problem without zero?

125300
179030
--------
304330

You have to be able to show, for example "zero tens." Now that's not to say you couldn't do it, calculations were done before zero came along, but it makes it a lot easier. Try it with Roman numerals or cuneiform wedges and see how long it takes.

[NB: I reserve the right to be completely wrong. ]

 

[matt grime]

and how many times don't you exclude 0 from things just because it won't work...

how often do you exclude things from the domain of a function because there is no sensible way ot extend that function to that domain without passing out of the intended codomain? Always.

 

[anti_crank]

ok... sure.. you can live in ignorance

This comment is best reserved for the peculiar character you see when you look in the mirror.

and don't care of the somewhat interesting fact that 0 doenst vehave as other integers/rational numbers...

All integers behave differently. Can you find me more than one integer x such that 2+x=3?

ALL the arithmatic operations are applicable on EVERY rational number, but not 0...

Nor do we require this. We want the real numbers to form a field, because fields have many useful properties.

and how many times don't you exclude 0 from things just because it won't work...

If you can quote me five distinct instances of that, I will buy you a Coke. Funny though that you didn't exclude it when you said 100%. What I suggest is that you think of zero as a placeholder for decimal notation; surely you can see the usefulness of that? Otherwise, read on.

Really, the problem is that you don't know how the natural numbers are formally defined. Try learning some set theory; you can start here. Look in particular at the axiom of infinity. Now this defines the natural numbers. Integers are formally defined as pairs of naturals mod. an equivalence relation; rationals are then defined as pairs of integers mod. another equivalence relation; the reals are constructed from Cauchy sequences of rationals etc. These constructions are done so that various properties can be formally proved, not because we like it so. Thinking of numbers as having some sort of correspondence to the 'real world' (by which I mean the view that the number one comes from 'one apple', the number two comes from 'two apples' and so on) is fine to do grade school arithmetic; it already fails in high school since there are no 'pi apples' and there never will be, and is utterly useless for formal study in university and beyond.

Did you know that there are spaces where zero can mean something totally different? For example, in the L² space of square integrable functions on the real line, take the function f defined such that f(x)=1 for x rational and f(x)=0 for x irrational; this function is then formally equivalent to zero. But to know that, you need to know a lot more mathematics than it seems you know.

I'll stop this discussion then, because people seem to not be interested in more than what their mathbook say to them, and seem to not be able to do some thinking of their own. No, because if you need to think you refuse by saying "that is philosophy". I bet none of you will never come up with something new in the mathematics...

And you will lose that bet. If this is the way you want to approach things, don't let the door hit you on the way out.

 

[HallsofIvy]

I think I can answer strid's question in exactly the way he wants-

0 is of absolutely no importance- to someone who is determined not to learn any mathematics at all! (I assume that sarcasm is acceptable to strid.)

"what makes zero more number than infinity? if zero is a number then infinity has to be as well...". No one has SAID that infininity is a number (in fact you are the only one who has mentioned infinity here). Yes, both 0 AND infinity are numbers- at least according to the the definition of "number" that I use (what definition of "number" are YOU using? Or do you even care about definitions?). 0 IS a "real number" and infinity is NOT because while 0 satisfies the definition of "real number" infinity does not.
(If you do not know the definition of "real number" then I would recommend you learn the definitions of things BEFORE you start debating them.)

 

[strid]

No one has SAID that infininity is a number (in fact you are the only one who has mentioned infinity here)... Yes, both 0 AND infinity are numbers

was thikning of not posting anymore here but this was to ridicolous...

First you say that no one has said that nfinity is a number (which i interpret as that you mean it isnt).. than you say infinity is a number... you seem confused...

you don't seem to know so much either... join the club! (note the sarcasm)

Infinity IS NOT a number...
Surprised to hear a Super Mentor say that with more than 4000 posts...

Infinity is not a number; it is the name for a concept.

 

[arildno]

You seemed to overlook the crucial part of Halls' sentence:

..- at least according to the the definition of "number" that I use (what definition of "number" are YOU using? Or do you even care about definitions?).

But "definitions" aren't something you bother learn about, is it?

 

[strid]

... he said "according to the definition of "number" that I use"... so... His definitions are completely wrong then... and the point is that he thinks that infinity is a nnumber, which it absolutely isnt..

 

[arildno]

Sure infinity can be a number.
It just depends on what number system you're talking about.

 

[anti_crank]

Infinity is not a number; it is the name for a concept.

Add the extended real number system to the list of things you're ignorant about.

First you say that no one has said that nfinity is a number (which i interpret as that you mean it isnt).. than you say infinity is a number... you seem confused...

you don't seem to know so much either...

HallsOfIvy was careful to distinguish between numbers according to some definition on one hand, which may include none, one, or many concepts of infinity, and the formal real numbers on the other, which do not include any such objects. This is exactly the kind of rigor that you don't seem to grasp. Now stop correcting those here who have PhD's in mathematics by arguing poorly understood grade school mathematics.

To be completely honest, no one could care less whether you want to accept zero or infinity as numbers or not, so do whatever you want. Mathematicians have agreed to accept those concepts as meaningful because they are useful; whether you want that functionality depends on the results you are interested in.

 

[strid]

infinity is NOT a number...

IF it is.. please tell me what the number you get when you take infinity + 1...

i can giive you 10 pages on the net saying that infinity is not a number...

 

[infidel]

If it's on the 'net, it must be true!

 

[Data]

Infinity is a number, in some number systems (the extended reals, for example).

It is not an element of any of the sets of whole numbers, natural numbers, integers, rationals, reals, or complex numbers, though. Really, talking about infinity at all in reference to these systems is meaningless. When we do so, we are appealing to the fact that we can extend our systems to include infinity as an element. We do not use these extended systems in most situations, though, because if we do then many of our operations need to be redefined (infinity does not work naturally with ANY "arithmetic" operation). It is simply a matter of convenience.

The fact that you have been taught that infinity is strictly not a number is irrelevant. Indeed, from your perspective it is probably an ill-defined concept. This does not make it so from the perspective of mathematics.

An identity element, e, of a nonempty set S with respect to a binary operation \langle \ , \ \rangle: S \times S \longrightarrow S is one such that

\langle x \ , \ e \rangle = \langle e \ , \ x \rangle = x

for every x in S.

With respect to the integers and multiplication, 1 is an identity element. In the exact same way, with respect to the integers and addition, 0 is an identity element.

The natural numbers, integers, rationals, reals, and complex numbers are mathematical contructs. Trying to put them in direct correspondence with things in the world in which you exist is wrong. Sometimes we are lucky, and can discover some such constructs that model the world in a sufficiently good manner. Such constructs are usually described as "natural" or "intuitive," but these are purely subjective terms.

Let's say we have a set E with a binary operation \cdot. In addition, assume that it does not have an identity element with respect to \cdot, so there is no element x such that

x\cdot y = y \cdot x = y

for every y in E.

We can then define an object e by

e \cdot y = y \cdot e = y,\; \mbox{and} \; e \cdot e = e

for every y in E. Then the set E \cup \{ e \} does have an identity element with respect to \cdot. If E is the whole numbers, and \cdot is +, then we can perform precisely these steps to get an additive identity. We just call this identity element "zero" or 0.

Simple mathematical construction.

In contrast, we could define an "infinity element," i, of a set S with respect to a binary operation \langle \ , \ \rangle: S \times S \longrightarrow S by

\langle i \ , \ x \rangle = \langle x \ , \ i \rangle = i

for every x in S.

Under this definition, we can look at the natural numbers and addition. Is there any element satisfying this definition? No. Can we define one? Certainly. Define \infty by

\infty + x = x + \infty = \infty, \; \mbox{and} \; \infty + \infty = \infty

for every natural x. So now, the set \mathbb{N} \cup \{ \infty \} does have an infinity element under our definition, and \infty is a number, ie. an element of the set.

 

[arildno]

infinity is NOT a number...

IF it is.. please tell me what the number you get when you take infinity + 1...

i can giive you 10 pages on the net saying that infinity is not a number...

What is the number system you're working with?

 

[matt grime]

I bet they say "infinity is not a real number", or that the real is implicit. Check your definitoins.

There is the extended real line which possesses plus and minus infinity. And then infinity plus one = infinity by continuity.

There is also the extended complex plane which has the point at infinity.

Then there are infinite cardinal numbers, and more than that there are the surreal and hyperreals that all have some notion of infinity being a useful number IN THAT SYSTEM.

 

[Palindrom]

Dear god, where did that discussion go...


strid: Like I've tried to tell you before, and don't take this the bad way, you relatively have no idea what you're talking about.
I won't try to take this discussion any further, because I feel you won't listen anyway.

Telos- I didn't phrase myself correctly. What I should have said is you can't explain complex Mathematical notions to someone non-academic who won't listen.
And just for kicks- suppose I was a random guy from the street. How would you connect a Galois Group to my everyday life?

 

[robert Ihnot]

\infty + x = x + \infty = \infty

Is everyone really sure of that? In the sense of an ordinal number?

The following idea is Cantor's: "Following the logical definition of w, Cantor further devised the concept of even larger sets. If you imagine w to be the order, or size, of the set {0, 1, ...} of all countable numbers, this set could not include w because w is Inf. Adding w to that set would produce a set 1 bigger than w, which Cantor denoted w + 1. It must be noted however that Cantor did not consider 1 + w to be the same as w + 1: the former meaning the set of one element, {0} + {0, 1, ...} [the Infinite Set] = {0, 1, ...}, the later meaing the set {0, 1, ...} + {w} = {0, 1, ..., w}. Thus we have the somewhat startling result that 1 + w = w but w + 1 > w." http://starship.python.net/crew/timehorse/new_math.html

 

[Data]

You are right, of course, if you are using Cantor's definitions. I wasn't.

My definition of an "infinite element" doesn't work out if you try to use it in some other examples anyways. But it does in the simple context that I needed it

 

[matt grime]

WEll, the arithmetic of the ordinals and the cardinal doesn't have to be the same. Something that the OP probably ouwld strenuously object to