The Importance of Zero: Uncovering its Significance

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The discussion centers on the significance of zero in mathematics, with participants debating its status as a number and its practical utility. Zero is recognized as the additive identity, meaning it plays a crucial role in algebra and mathematical structures, allowing for operations that define other numbers. The conversation also touches on philosophical questions about existence and the nature of numbers, comparing zero to infinity, which is treated as a concept rather than a number. Participants argue that without zero, concepts like negative numbers and certain mathematical operations would become problematic. Ultimately, the importance of zero is tied to its foundational role in mathematics, enabling a coherent framework for understanding numbers and their relationships.
  • #61
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)
 
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  • #62
strid said:
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.
 
  • #63
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.
 
  • #64
strid said:
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...
 
  • #65
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.
 
  • #66
Alkatran said:
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 :))...
 
  • #67
strid said:
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!).
 
  • #68
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.
 
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  • #69
strid said:
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 ?
 
  • #70
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.
 
  • #71
strid said:
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 :))...
On the contrary, we understand you perfectly well.
You are clinging to your own personal fantasies as to what numbers OUGHT to be, and, because fantasies are fuzzy, warm and cozy, you want to live with them, rather than learn how to think by means of rigourous logical systems, which you fear because they seem strange, cold and hard to you.
You are locked in emotionalism, that's all there is to it.
It is not difficult to understand you at all.
After all, your condition is quite prevalent in the human race..

And, you seem to have missed out something: Everyone here agrees that infinity is NOT, for examples: a natural number, integer, rational number or real number.
The fact that there are lots of number systems in which infinity cannot be regarded as a number does not make it impossible to comstruct legitimate number systems in which infinity IS a number.
It is really not anything more special than that "most" fractions cannot be considered as natural numbers, but ARE rational and real numbers.
 
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  • #72
Gokul43201 said:
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 ?

sorry.. i mistyped... of course i meant either less than -300' C or less than 0K...


Will try to answer on what I'm critized on right now...

My definition on number (in this context) is a quantity... It can be anything quantitive including lengths and other such stuff...

My first reason to think that 0 was not a "number" was that I saw it as much number as infinity... It is sort of like that the numbers 0,0000...1 to 10^9999... are possible to exist in a totally different way than zero an infinity...
Its like that zero is infinitely small while infinity is infinitely big..,,

And the stuff with that you can't divide with zero, is that you can divide by all other "numbers"... so the fact that you can't divide vy zero makes it somewaht different from other numbers...
 
  • #73
In that case you're definition of "numbers" is completely different from any mathematical one. So you can quit worrying about mathematics. The problem isn't mathematics it is yours.

Incidentally, water freezes at 0 degrees C, so zero exists there as a measurement.
 
  • #74
strid said:
so the fact that you can't divide vy zero makes it somewaht different from other numbers...
This is only true if your system does not define division by zero. For example, taking the square root of a number that is not a perfect square is impossible in the rationals, making those numbers different from other numbers. You claim we should then take these so-called "numbers" out of the system, instead of finding a meaningful extension of our system. The latter choice brings new vistas of mathematics, while the former choice is a step backwards. Your personal problem with zero is echoed by others' problems with other aspects of other systems. Some may not want any numbers other than 1, because it makes no sense to define a new number other than a whole object. You may argue against this, but I'm sure you can see that your arguments will be just as futile as ours are to your belief.
 
  • #75
hypermorphism said:
This is only true if your system does not define division by zero.

is there any system where it is defined??
 
  • #76
Dear God do you not read the posts here? The extended real numbers, the extended complex plane, both allow you to define 1/0 (though nto 0/0 for obvious issues with continuity).
 
  • #77
matt grime said:
Dear God do you not read the posts here? The extended real numbers, the extended complex plane, both allow you to define 1/0 (though nto 0/0 for obvious issues with continuity).

Hold on, they do? Don't you need limits for it to make any sense?
 
  • #78
Nope. However, you have to be careful with them; ordinary arithmetical facts like x + 1 != x don't always hold in these systems.
 
  • #79
Hurkyl said:
Nope. However, you have to be careful with them; ordinary arithmetical facts like x + 1 != x don't always hold in these systems.

I'm assuming that you're talking about +- infinity (or in the case of the complexe numbers, complexe infinity)?
 
  • #80
I think the OP may have just read this book, which over-hypes the importance of zero from a historical perspective.
 
  • #81
ok... if now 1/0 is defined... than what is the differnce between 1/0 and 2/0? are they equal or what?
 
  • #82
If \frac{1}{0}=\infty, then you could say \frac{2}{0}=\frac{2*1}{2*0}=\frac{2}{2}*\frac{1}{0}=1*\infty=\infty

Heh.
 
  • #83
Moo Of Doom said:
If \frac{1}{0}=\infty, then you could say \frac{2}{0}=\frac{2*1}{2*0}=\frac{2}{2}*\frac{1}{0}=1*\infty=\infty

Heh.


yeah right...

you can also say that

1/0 = inf.
2/0= 2 *(1/0) =2*inf.


or...
2/0= 2/ (4*0) =0,5* (1/0)= 0,5 * inf.

As you see you can quite many answers... :)
 
  • #84
Again, Strid, you have not in stated in which system you are talking about. Why don't you actually do that?

In Cu{\infty} 1/0=2/0=\infty.

This is "by continuity".

You do understand that things in mathematis essentially follow from the definitions and not your real life intuition?
 
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  • #85
strid said:
yeah right...

you can also say that

1/0 = inf.
2/0= 2 *(1/0) =2*inf.


or...
2/0= 2/ (4*0) =0,5* (1/0)= 0,5 * inf.

As you see you can quite many answers... :)
And why don't you think that we may have 2*inf=inf and 0.5*inf=inf?
 
  • #86
you seem to missed the sarcasm again.. i answered Moo of Doom... read his post before criticing mine...
 
  • #87
"criticing"... what a delightful new word in English!
Where did you find it?
 
  • #88
Strid, why don't you sit down and write out the lits of rules that your "numbers" must satisfy. Then attempt to show if there is or isn#t a model of this system.

Because the "numbers" in mathematics are axiomatic constructs. Stop trying to use your "intution" on them. We have axioms, we know they are not self contradictory since we can produce a model of them. And we can deduce results about them. Notice, we deduce things, we don't make wild and unmotivated guesses that we insiste must be true even after it has been carefully explained to us why this guess is wrong.
 
  • #89
Just to help you along a bit with that list of rules we're waiting for, strid:

Do you want the following rules to apply to your numbers:
1) Whenever I add two numbers, I'll get a number back.
2) Whenever I multiply two numbers, I'll get a number back.

Will your system have these two rules, for example?
 
  • #90
Also, here are two examples of lists of rules (axioms) that you are (hopefully) already familiar with:
A ring (specifically, a ring which is an integral domain), a model of which is the set of integers under addition and multiplication.
A field, a model of which is the real numbers under addition and multiplication.
 

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