The Concept of Zero

[HHayashi]

Hello, first time posting here. I had an idea that was irking me and I would like as much input as possible. I'll just copy-paste something that I put on another forum (not this site).

 

I'm sure some people will be disgusted at some of the logic, but I just want clarification. I do realize that it's all hypothesis with no real mathematical proof, I simply don't have enough knowledge in mathematics to even get close to trying. I'm all ears though, so if there is some mathematical formula to shed some light then I'll try my best to understand it.

 

The basic question I want answered is, "Is Zero a real number?" (insert gag reflex)

 

My train of thought first came from why you can't divide by zero. You can divide something by a number that infinitely approaches zero and it approaches infinity when you look at any graph. As in, x/ε = ±∞. The same can be said for the inverse, so x/∞=ε.

 

I remember the first thing I learned in Calculus was that any number that infinitely approaches a number can be considered the same, as in .999999999......= 1. So why doesn't it work with zero? I just hear people say, "well, zero has a few exceptions because it's an identity element". As far as I can recall, 1 is also an identity element and it's keeping itself in line just fine. So then I thought, "Is Zero a real number?"

 

There's actually another "number" in mathematics where it infinitely approaches the number yet it can't be considered the same, and that is in fact "∞". Why is something infinitely large not considered a real number, yet something infinitely small is considered a real number?

 

So then I thought, "Maybe zero is a concept and it's not a value you can actually reach, just like ∞". This makes me think that there's a good possibility that zero shares more traits with infinity than a real number. This means that 5+0, 5-0, and 5*0 are all something you simply can't do, just like infinity. If you look at it another way, adding, subtracting, and multiplying "nothing(zero)" might also be impossible since it isn't actually a value. By this logic, every instance of zero in current mathematics was actually always ε, so 5+ε=5, 5-ε=5, and 5*ε=ε. I pondered the thought, and found myself deducing that we wouldn't notice any difference whatsoever. I did some brief research on infinitesimals, and I'm pretty sure this is just a different way of approaching non-standard analysis. It was really brief research though, and I don't have any idea where to start.

 

So, why is it important if it doesn't make a difference? To be honest, I don't know. This was the same treatment that non-standard analysis got, in that it was just a different way of achieving the same answer. However, it might make a difference somewhere that I'm not aware of though, like quantum mechanics or something. Maybe non-standard analysis can get somewhere that normal calculus can't because of that little difference.

 

Of course, I could be completely and utterly wrong somewhere. Criticism much appreciated.

 

P.S. I'm not trying to prove anything with this post, I'm looking for something that can clarify it a little more. I realize that I'm just assuming zero is more similar to ∞ than a real number without any proof whatsoever.

 

**

 

Part of the response I got in another forum was to think of numbers in most systems as something of measurable value. Well, I dunno about you but I cannot measure 0 centimeters.

 

One thing particularly intrigued me. The same fellow also said that limits are a great at explaining what happens when something gets infinitely close to something, but it doesn't actually become the same thing. Well, this may be the single greatest proof that zero is not an attainable value, and as such is not a real number.

 

0.999999....=1, but it doesn't actually ever get there. This means that the difference between two numbers cannot actually ever be zero, just infinitely small. This seems to be the case for proof of 0.999.....=1 using infinite geometric sums, since infinity is also not an attainable value.

[John Cuthber]

Zero centigrade is the temperature half way between +1 and -1. On a cold night the temperature will drop through zero, for an instant it will be zero. Zero exists.

[studiot]

Well it depends upon what you want your number system to do; not all number systems incorporate a zero.

 

Fo example the the counting or natural numbers 1,2 3, .....

 

have no zero, as you have observed with your measuring comment.

 

However mathematicians like closed systems this means that we can use a symbol to represent a particular property of all the elements of such a system.

 

Zero is that element which allows us to say

 

For any three numbers a and b and c in our numbering system

 

we can find a b such that (a+b) = c

 

Now if a = c then we must find a b such that a + b = a and zero is this number for c.

[John Cuthber]

Oh, btw, re.

"0.999999....=1, but it doesn't actually ever get there. This means that the difference between two numbers cannot actually ever be zero, just infinitely small."

Just plain wrong.

0.9 recurring is exactly 1 and the difference is zero: the proof is on the web if you want to look.

[doG]

1-1=0 ... simple enough

 

On a philosophical note I recommend Zero: The Biography of a Dangerous Idea by Charles Seife. I quite enjoyed it.

[imatfaal]

1-1=0 ... simple enough

 

On a philosophical note I recommend Zero: The Biography of a Dangerous Idea by Charles Seife. I quite enjoyed it.

 

I'll let you know what I think - only a few quid on Amazon, cost me only a bit more than a sunday paper. I love second hand books on the marketplace - can get quite compulsive when decent books are so cheap

[Bignose]

Is Zero an attainable value?

 

I own zero Lamborghini automobiles. Seems attainable to me....

[Phi for All]

I own zero Lamborghini automobiles. Seems attainable to me....

The new models are out. I was going to get one but didn't, so that's how many I have, too.