The Importance of Zero: Uncovering its Significance

[strid]

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?

 

[matt grime]

Before you talk about existence or non-existence of zero would you care to explain what mathematical objects you think do exist and in what sense. Moreover, why are these other things you trust not also inventions "just to make stuff work"?

 

[strid]

the thought I'm playing with for the moment is that every rational number has its origin in 1. So by multiplying with 2 you get 2. by multiplying with 500 you get 500. And by dividing by 2 you get 0,5. Dividing by 500 give you 0,002.

So the bigger number you mulitply with the closer you approach "infinity"... but you can never "reach" it... The same way, the bigger number you divide it by, the closer you come to zero, but you can never "reach" it... so just as infinity is not a "number", zero is neither a number...

EDIT: to rephrase my question... what makes zero som important in maths.. what problems did we have before we discovered the zero?

 

[Zurtex]

All rational numbers come from integers, where a is some integer, 0 is defined by: a + 0 = a. That's pretty much all there is to it.

In integers, rational, real and complex numbers a/0 is not a number because dividing by 0 is not defined.

 

[strid]

but what is then the profit of the zero? why was the discovery of zero so revolutionary??

the 0 doesn't help us a lot in a+0=a
why not just skip the 0.. its useless (at least in this case)...

so can anyone come up with a place where the zero is good,,..

 

[matt grime]

It is the cardinal number of the empty set, it is the additive identity of the reals, etc. It is therefore important in field theory and thus all of algebra and without it and the ideas contained in the notion of identities and inverses there would be practically no modern mathematics and physics. And you'be still not said why you think 1 "exists"

 

[strid]

"It is the cardinal number of the empty set, it is the additive identity of the reals, etc. "
didnt understnad anything of that... could you explain one of those instead of just mentionening it..

if i have an apple i name the quantity 1... so if i have two apples i can i can do 1*2 and get 2 apples... so in that sense 1 exists... but saying that you have 0 apples when you have nothing is not the same thing... because 0 is nothing...

 

[SpaceTiger]

if i have an apple i name the quantity 1... so if i have two apples i can i can do 1*2 and get 2 apples... so in that sense 1 exists... but saying that you have 0 apples when you have nothing is not the same thing... because 0 is nothing...

This sounds more like a philosophical question. Is nothing something? Who knows. "0" is certainly something because we give it meaning and use it as an object, but that which it represents may actually not be something. The same would go for the word "nothing". It exists and has definition, but is a pointer to nothing.

 

[strid]

then infinity?

there are inifinte many points in any line... does that make infinity a number?

if 0 is nothing infinty is sort of everything... they have both a mathematical value but none of them, according to me, are numbers.

 

[Zurtex]

then infinity?

there are inifinte many points in any line... does that make infinity a number?

if 0 is nothing infinty is sort of everything... they have both a mathematical value but none of them, according to me, are numbers.

Give us how you define a number then?

In mathematics numbers are constructed from sets.

1 is the multiplicative identity as: 1*a=a
0 is the additive identity as: 0 + a = a

Both are highly useful.

 

[SpaceTiger]

then infinity?

there are inifinte many points in any line... does that make infinity a number?

Meh, I think the definition of "number" should be left to the mathematicians, seeing that it's a mathematical concept.

 

[strid]

I don't see how a+0=a can be useful... its the same thing as a=a

i'vent thought through this very much but what criterias a number needs to have is at least that they can cope with the arithemtics which neither zero nor infinty does...

a/0 is undifined as well as a/infinity...

 

[arildno]

The basic reason why you don't get this, strid, is that you mix your own silly, private fantasies of numbers, and what operations you might do with them into the properly defined area of mathematics.
I have a suggestion for you:
1. Start to regard your own thoughts as the dumb results born of ignorance.
2. Start learning maths properly.

 

[strid]

arildno, you arguments are magnificient... are you a lawyer? I'm totally convinced that you are right and I've surely leartn something new...
(in cas you don't get it, I'm sarcastic)

However...

what makes zero more number than infinity? if zero is a number then infinity has to be as well... or, as i am claiming, none are...

 

[Giuseppe]

Agreed

I agree. Just accept the number. What did it ever do to you? In any system, like one with mathematical relations has to define what "something" is, on the contrary, it must also define what "nothing" is. Don't major in philosophy.

 

[arildno]

"what makes zero more number than infinity? if zero is a number then infinity has to be as well... or, as i am claiming, none are..."
Why?
How should I know what idiotic personal thought-process has lead you to this result?
It certainly hasn't anything to do with maths or logic.

Since you show an active unwillingness of actually learning anything, here's a new suggestion:
Keep your private mental jerk-offs to yourself.

 

[Giuseppe]

Continued

Its only more practical to treat zero more like a number. Do we ever say "I have infinity apples?" It makes sense to say that I have zero apples. Plus, we use infinity usually to see how something behaves (something like a series) as it approaches infinity. We even do the same thing with zero (at least not with series)

 

[Giuseppe]

Nowhere

This isn't going anywhere

 

[strid]

"what makes zero more number than infinity? if zero is a number then infinity has to be as well... or, as i am claiming, none are..."
Why?

re-read what I've written.. or i can rewrite...

1. if we start from 1... the higher number i multiply with, the closer i get to infinity... the higehr number i divide 1 with, the closer i get to zero...

so zero is the result of an infinite serie as well...

also...
you can't use the arithmatcis operations on infinty very well... just the same thing with zero...

how come you can have a/b for any "number" of b except 0??! might it be that 0 isn't a "number"??

Giuseppe
No, it wouldn't make sense to say that you have infinite apples as there arent so many apples...
but.. you could say that there are infinite many points in a line.. does that make infinity a number?

 

[Zurtex]

I don't see how a+0=a can be useful... its the same thing as a=a

i'vent thought through this very much but what criterias a number needs to have is at least that they can cope with the arithemtics which neither zero nor infinty does...

a/0 is undifined as well as a/infinity...

a + 0 = a

Is very useful because it established there exists an element in the set in question such that when you add it to any other element in the set it is equal to the other element in the set. This is not true for most sets of numbers.

Consider a/107 is not defined in integers for most integers a.

re-read what I've written.. or i can rewrite...

1. if we start from 1... the higher number i multiply with, the closer i get to infinity... the higehr number i divide 1 with, the closer i get to zero...

so zero is the result of an infinite serie as well...

also...
you can't use the arithmatcis operations on infinty very well... just the same thing with zero...

how come you can have a/b for any "number" of b except 0??! might it be that 0 isn't a "number"??

Giuseppe
No, it wouldn't make sense to say that you have infinite apples as there arent so many apples...
but.. you could say that there are infinite many points in a line.. does that make infinity a number?

All numbers can be wrote out as infinite series in real numbers, that is in fact one of the definitions of real numbers.

a/b represents all rational numbers, for a in the integers and for b in the natural numbers, that works perfectly.

 

[Palindrom]

First of all, infinity cannot be an object in an algebraic structure as we know it- i.e., there are no operations like multiplication and addition with infinity.
It's just a concept- for this discussion, anyway.

Zero, on the other hand, is a number. It is part of every basic algebraic structure (a group) by definition. Zero isn't only a+0=a, but also for all a you have a*0=0- so now it's already interacting, isn't it?

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.

Say you have 6 apples. Now, I'm taking 2 of them from you. How many apples do you have left? That's right, 4.
Now suppose I'm taking 1 more. You are now left with 3.
You might notice that while I'm taking apples from you, you are left with what you call a certain "number" of apples.
Now say I'll take your last 3 apples. How many are you left with?
Hum. None? So what is none? Is it also a number?
And what if I want to take even more apples now? Why can't I? It works in the bank, doesn't it? When you owe someone apples, how many apples do you have? Still none? So when I give you an apple, you'll have 1? But you owe some apples to people, so you'll still have none... so now something's wrong, are there different kinds of "none"? Or do you accept the concept of a negative number of apples?

Are you sick of damned apples yet?

To ignore the existence of "0" would be to ignore the existence of negative numbers, and eventually you'll start seing things like "No apples + 1 apple= No apples".
So is 1 actually none? So are there no positive numbers either?

There's always something you don't understand... so ask to know, not to make people know. It only makes people... mad.

 

[Giuseppe]

BRAVO! Palindrom!

 

[arildno]

Consider first the natural numbers 1,2,3.. and so on.
Now, consider PAIRS of such natural numbers, forexamp le: (3,5), (7,1), (3,3).

Now,the INTEGERS neatly divides up this set of pairs by recognizing a common property among subsets of the whole set of pairs.
For example, the INTEGER "1" neatly represents member pairs of the form (2,1),(3,2),(4,3) and so on.
The INTEGER "-1" neatly represents member pair like (1,2),(2,3),(3,4) and so on.

Now, you can figure out if the INTEGER "0" can neatly represent some such subset of member pairs.

 

[strid]

palindrom... your arguments are the first constructive arguments that have any meaning I've been given...

your arguments are well noted, but there is still a thin gthat is a thorn in the eye...

why is a/0 undefined... it is defined for any other number... and zurtex, a/107 is always defined.. you can give it as a freaction if you want and in decimals you can give a fairly precis answer but with a/0 we have no idea what the answer is (if ther now is an answer)...

I'm not trying to be stubborn and refuse to accept the zero... I'm just trying to get the importance of zero as well as why zero is defined as a number, because there ARE problems with that...

 

[Palindrom]

Yout question shouldn't be "why is a/0 undefined?", but rather "can we define at all a/0?"

Well, within the other definitions we've made, we can't. It can be easily shown.
Suppose you could, and let a be a non-zero number. You would then have:

0*(a/0)=a

But it is common knowledge (and the first thing you prove after defining a ring), that for all b, 0*b=0. (because take b=a/0, which is well defined by our assumption, and we got 0*b=a which is non-zero)
And that's a contradiction. We can't have that. If we have contradictions, All of what we do is invalid.

 

[strid]

"And that's a contradiction. We can't have that. If we have contradictions, All of what we do is invalid."

thats why I'm questioning zero as a number.. it is like the infinity stuff.. all you do gets invalid.

isnt it a reasonble criteria for a "number", that they can handle all the arithatic opertations?

 

[matt grime]

No that isn't reasonable at all, since we can demonstrate that declaring such is inconsistent.

Are you aware of the distiction between the natural numbers, the integers, the rational numbers, the real numbers and the complex numbers? The names of these indicate the old fashioned prejudice that your exhibiting by the way. It's depressing that it's these pointless "well, I don't think it should be so" threads always get the most attention.

Note that integers do not possesses all divisors, and rationals do not even posses roots of positive numbers, and the reals do not have roots of negative numbers, if yo'ure going to start talking abhout operations

 

[arildno]

"And that's a contradiction. We can't have that. If we have contradictions, All of what we do is invalid."

thats why I'm questioning zero as a number.. it is like the infinity stuff.. all you do gets invalid.

isnt it a reasonble criteria for a "number", that they can handle all the arithatic opertations?

I have already told you why zero can readily be accepted as an integer along with the other ones.
The fact that you simply are unwilling to learn is just more and more apparent.

 

[Palindrom]

Nothing gets invalid, as long as you are carefull. You simply can't define the multiplicative inverse of 0, that's all. That's the price you have to pay for your definitions.

You ask "isnt it a reasonble criteria?".
Is it?
I could give you objects that satisfy that criteria, and you would still probably be the first to say "that's definitely not a number".

On the other hand, once, people didn't know about rational numbers. Their entire world was Z, the whole numbers.
But in Z, you can't divide by 3. It's not always well defined. 2/3 isn't a whole number, and these people didn't yet accept the notion of rational numbers. So isn't 2/3 a number?

Now let me ask you another question: is the square root of minus 1 a number? Does it even exist?

Or should I ask- can it be defined?

 

[strid]

Note that integers do not possesses all divisors, and rationals do not even posses roots of positive numbers, and the reals do not have roots of negative numbers, if yo'ure going to start talking abhout operations

I said arithmatic opeations.. the root is not arithmatic, or?
could you please clarify what you meant by "integers do not possesses all divisors"

i know that we can't get every number by fractions.. such as PI... but.. we can locate the number pi somewhat arounf 3.1416.. but 1/0 we have no idea where it is on the time line...

and the stuff you said about integers, comples numbers, rational numbers etc... how do you classify 0? is it a rational number?

EDIT: palindrom... The square root of -1, i, is indeed a number, but a complex number... it is a number outside the "natural" number line and hence a very special sort of number. It isn't a quantity, not can be a length of anything so... once again... what sort of number is 0? I would accept and agree with that it is a number but not a rational...

 

[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.