# Number infinitely close to one but not one

1. Jun 19, 2013

### firefly431

How would one notate a number (any algebraic entity) that is infinitely close to one but not one? I know that 0.999...=1, and thus cannot be *not* one, so what would be a suitable name for this number?

Last edited: Jun 19, 2013
2. Jun 19, 2013

### BenjaminTR

There is no real number that is infinitely close to 1 yet not identical to 1. Thus, one must use a different structure, such as the hyperreal numbers. The interesting thing about switching to hyperreals is that using the same decimal numbering convention, where

.999... = 9/10 + 9/100 + 9/1000 ...,

Then 0.999... does name a number infinitely close to 1 but less than 1. I know I read a paper on this, and I will track down the reference soon.

If you do not go this route, the name would probably have to be 1 - epsilon, where epsilon is an infinitesimal.

3. Jun 19, 2013

### firefly431

Is there any special notation for writing this number? I don't know much about hyperreal numbers.

4. Jun 19, 2013

### BenjaminTR

This is the reference I was talking about: http://arxiv.org/abs/0811.0164

I don't know if there is any notational convention for specifying that you are using the hyperreal number system.

5. Jun 19, 2013

### glappkaeft

The same as for any other type of number, in this case you could write, for example, 'x is an element of *R'.

The hard part would be to learn all the theorems and rules that differ slightly between the hyperreals and the real numbers.

Since hyperreals are almost never used and one would like to express something similar it would probably be better to continue using the reals and write something like 1+x, where x is a real number arbitrarily close to 0 (but NOT 0). This is of course not a unique number.

Last edited: Jun 19, 2013
6. Jun 19, 2013

### HallsofIvy

Staff Emeritus
Do you understand that everyone is saying "as long as you are working in the real number system, there is no name for such a number, because there is NO such number!"

7. Jun 19, 2013

### Staff: Mentor

No, 0.999... and 1 are the same number, so neither is less than the other. The ellipsis, ..., is significant, and indicates that the 9s repeat forever.

8. Jun 19, 2013

### micromass

Staff Emeritus
He is talking about the hyperreal number system. In that number system, what he is saying is absolutely correct.

9. Jun 19, 2013

### Integral

Staff Emeritus
Let's stick to the real number system. The smallest non zero real number does not exist. So how do you name something that does not exist? Inf ℝ would be a possibility.

How ever this is a very meaningful question to ask about your computer, what is the smallest number it can represent? Perhaps the computer guys have a name for it. A quick google turned up RealMin in Matlab. I think you can do some more searching and find others. Take your pick.

10. Jun 19, 2013

### ahgamer

Call it anything you like

What matters is that there be an intelligible definition. Maybe something like ε such that for any ε-prime where ε ≠ ε-prime

1 - ε < 1 - ε-prime

which is to say that the magic number is always less than ε-prime. That would make it the closest number to 1.

11. Jun 20, 2013

### ahgamer

Well, the above needs some fixing. The closest number to 1 is ε where ε ≠ 1 and where for any ε-prime such that ε-prime ≠ ε and ε-prime ≠1

abs(1 - ε) < abs(1 - ε-prime)

12. Jun 20, 2013

### Staff: Mentor

I stand corrected.