# Actual infinitesimal, actual infinity

1. Jul 10, 2009

### Pjpic

Is there an actual infinitesimal in the way that there is an actual infinity. Or would zero fill that role.

2. Jul 10, 2009

### Office_Shredder

Staff Emeritus
You can define a number system by adding an additional symbol $\epsilon$ and defining it by $\epsilon ^2 = 0$ and then taking the set of all $a+b \epsilon$ where a and b are real numbers. You can add and multiply like normal using distributivity and commutativity. But in the standard set of real numbers there is no infinitesimal, just like there is no actual infinity

3. Jul 10, 2009

### HallsofIvy

What do YOU mean by "actual infinity"? "Non-standard analysis" uses the "hyper-real numbers" with infinitesmals. But, as Office Shredder said, there is no "actual infinitesmal" just as there is no "actual infinity".

4. Jul 10, 2009

### CRGreathouse

I thought 0 was an infinitesimal.

5. Jul 10, 2009

### HallsofIvy

No, it isn't.

"In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, hence not zero size, but so small that it cannot be distinguished from zero by any available means."

" number system is said to be Archimedean if it contains no infinite or infinitesimal members."
and, of course, the real numbers are Archimedan.

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

6. Jul 10, 2009

### CRGreathouse

I'm accustomed to the definition "a system is Archimedian iff the only infinitesimal it contains is zero".

7. Jul 16, 2009

### Pjpic

8. Jul 16, 2009

### Tac-Tics

If by infinity, you mean "a number which no other is greater", then the real numbers contain no infinities.

If by infinitesimal, you mean "a nonzero number which is less in magnitude than all others", then again, there are no infinitesimals in the reals.

9. Jul 16, 2009

### HallsofIvy

The reason I asked was that your original post (which I have quoted here) implied that there exists an "actual infinity". There does not- not in the real numbers. There are many different ways to define both "infinity" and "infintesmal" in other systems.

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