The Importance of Zero: Uncovering its Significance

[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

 

[Hurkyl]

Or to throw something else into the mix, if H is an infinite hyperreal number, then H + 1 is simply a different, infinite, hyperreal number whose value is one more than H. (IOW, (H+1) - H = 1)

 

[HallsofIvy]

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...

Not so much confused as typistically inept- I meant to type "no one has said (in this thread) that infinity is NOT a number- except you."

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

That's a club I'm a charter member of!

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.

And you STILL haven't said what you think a number is! I was under the impression that ALL numbers are concepts. As I said, infinity is not a real number: i.e. it is not a member of the set of real numbers, defined for example by Dedekind cuts, or equivalence classes of sequences of rational numbers, etc. There are a number of different "infinities" all of which are "numbers" in the general sense- anything that is in one of the various systems that are considered "sets of numbers". If you don't like that general sense, please tell us what definition of number you are using.

 

[matt grime]

From what we can gather he wishes for numbers to be the "things" that have all arithmetic operations defined on them, thus necessarily if we accept any "number" exists so must zero (n-n) and so must n/0. Thus whatever he thinks numbers are he must necessarily accpet 0 and 1/0 are such. Despite being adamant that one isn't and one ought not to be.

 

[eNathan]

I know that the "discovery" (rather invention) of the number zero was revolutionary and is seen as VERY important...

I've always had some suspicion to the zero by some unknown reason... I decided some weeks ago to figure out what it is that is wrong with the zero...

So could someone please tell me in what ways the zero is SO very important...

What I've thought of yet is that the zero doesn't exist in reality but is just an invention to make stuff work.. but what?

What if you had nothing, do you have 1 or -1? none...

 

[Alkatran]

As stated before:
Let's say I grant you that any number that has an arithmatic operation that loads solution which doesn't exist will not exist
0 doesn't exist.
2 - 2 leads to 0, which doesn't exist, therefore 2 doesn't exist
therefore ALL numbers don't exist

Also:
10 = 1*10^1 + 0*10^0 but 0 doesn't exist, so 10 doesn't exist. Well damn...


THAT'S why 0 exists.

 

[strid]

As stated before:
Let's say I grant you that any number that has an arithmatic operation that loads solution which doesn't exist will not exist
0 doesn't exist.
2 - 2 leads to 0, which doesn't exist, therefore 2 doesn't exist
therefore ALL numbers don't exist

Also:
10 = 1*10^1 + 0*10^0 but 0 doesn't exist, so 10 doesn't exist. Well damn...


THAT'S why 0 exists.

ehm... there is no ogic in that...

why shouldn't 2 exist just because 2-2 equals nothing? it is as sayig that if i have 2 apples, and I take away 2 apples there are none left, hence there isn't anything such as apples...the same goes for the 10 stuff

I've been totallly misinterpreted in this topic, which might partly be because of my unclear statements, but I still insist on the fact that the infinity is not a number but a concept. My point from the beginning was that 0 is as much number as infinity, and if now you guys are saying that infinity IS a number than, for you 0 is of course a number as well... but for those of us that think that infinity is not a number (there are many of us) the zero becomes quite interesting...

I might fbe criticesed for this analogy but it is sort of like this:
There isn't a number infinity just as there isn't a temperature less than 300K. It just doesn't exist (how we now may define exist ...

 

[Data]

I've been totallly misinterpreted in this topic, which might partly be because of my unclear statements, but I still insist on the fact that the infinity is not a number but a concept. My point from the beginning was that 0 is as much number as infinity, and if now you guys are saying that infinity IS a number than, for you 0 is of course a number as well... but for those of us that think that infinity is not a number (there are many of us) the zero becomes quite interesting...

No, you just don't seem to understand how mathematics works. Read my post on the last page if you want to see a mathematical basis for what zero is (from a certain perspective, of course; it is certainly not the only way to approach the problem!).

 

[matt grime]

Strid, at no point have you ever said what you think a number is. We have all carefully qualified what we're talking about, and you have not.

Nor have you been able to explani why 0 isn't one of these numbers. But then you can't explain what a nubmer is so that isn't surprising. The best we've come up with is that it isn't a nubmer because 1/0 doesn't exist in the Reals (or whatever system you're using). So?

This is the difference between doing mathematics, and waffling on about numbers being temperatures and stuff like that.

I take my complex numbers to be the one point compactification of the plane - it makes complex analysis so much nicer to write out - and that has a point at infinity.

 

[Gokul43201]

I might fbe criticesed for this analogy but it is sort of like this:
There isn't a number infinity just as there isn't a temperature less than 300K. It just doesn't exist (how we now may define exist ...

You opened the door : the temperature in the room where I'm typing this right now is less than 300K (it is ~295K).

Now let's get you to define "number", wot ?

 

[Zurtex]

In the set of natural numbers, 0 does not exist. This causes problems if we want to use this, we would struggle to define how many apples there are in a bowel consisting purely of bananas, that is its practical importance.

Mathematically, the additive identity plays a greater importance, for example once we build up our set of axioms of the real numbers into theorems we can such results as, if:

ab = 0

then

a = 0

or

b = 0

or

a and b = 0

This is highly useful and allows us to solve many equations. By the properties of real numbers, 0 is a real number. I would highly suggest you look up what real numbers are because I have a strong feeling you are not aware of this:

http://en.wikipedia.org/wiki/Real_Numbers

Real numbers are not something mathematicians pull out of thin air, they are very well constructed. You may make your own set of numbers that does not include 0, but out of all sub sets of real numbers an uncountable amount of them don't include 0, that is not that important.

However I would gladly like to see you design a workable and practical number system without ever using 0, I would be very impressed if you can construct something as or more useful than what we have.