# Infinity: a concept or number?

1. May 12, 2012

### revo74

Is infinity a mathematical concept or a number? Please elaborate. Is there any debate over this topic or is there a consensus amongst academia?

2. May 12, 2012

### micromass

Last edited by a moderator: May 6, 2017
3. May 12, 2012

### HallsofIvy

We can't give a complete answer to that until you define what you mean by "number".

If you are referring to the ordinary "real number system" (and its subsets) then, no, "infinity" is not a number.

There are, however, ways of extending the real number system, one of which adds 'infinity" and "negative infinity" as numbers. Caution- the usual laws of arithmetic do not generally hold in such a system- you still can't say "infinity - infinity= 0".

4. May 12, 2012

### uperkurk

Infinity is never classes as a number. Take for example

Infinity - Infinity = ?

You can't add, subtract, divide or multiply by infinity.

5. May 12, 2012

### Whovian

Did you read HallsofIvy's response? It's classified as a number to which the normal rules of arithmetic don't apply sometimes.

6. May 12, 2012

### micromass

You can do $\infty + \infty$ if you want to. So you can surely add them. But you can't subtract them (that is: you can't add infinity and negative infinity).

7. May 12, 2012

### revo74

Didn't Hilbert and Cantor believe infinity was a number?

If you extended the real number system to include infinity then what isn't it indeed classified as a number?

8. May 12, 2012

### micromass

Got any citation for that?

Because you extend the real numbers, which means it wasn't in the real numbers to begin with. Infinity is, by definition, not a real number. It is an extended real number though. It's not a real number for good reason, because not all the usual operations work with infinity.

9. May 12, 2012

### SteveL27

I'd like to know your answer a simpler question. Is 6 a concept or a number? Isn't a number just a concept?

What I mean is, there is no 6 in the physical world. There are six apples, six days, six sandwiches. But the number 6, in isolation, is just an abstract concept in our minds.

So how is 6 any different than infinity? Aren't both just mathematical concepts?

10. May 13, 2012

### uperkurk

Extending the number system to include infinity would ultimately give infinity a value. Infinity does not have any direct value so it is not a number.

In case you're not sure, open up MatLab or any kind of programming compiler and you'll see that when you try to perform any calcuation using the word infinity you'll get an error or -0

11. May 13, 2012

### Mandlebra

If you are interested in doing some algebra with ω (the "smallest" infinity), you should read up on Conway's surreal numbers (often also referred to as games due to how they arose during the study of go and similar games). However, the surreals are not for the mathematically loose as their definition and doing proofs can be quite tedious!

The reason I mention the surreals is that in this system the expression ω and ω-1 are different beasts!

12. May 13, 2012

### jreelawg

I say it's not a number because it's not quantifiable.

13. May 13, 2012

### SteveL27

How about pi = 3.14159265... continuing forever. Is it quantifiable? Is it a number?

How about the imaginary unit $i$? Is it quantifiable? Is it a number?

Just for comparison, the estimated number of atoms in the universe is 1080, way less than just 101010.

Is it quantifiable? How many is it? Could anyone count that high? Is it a number?

How about the integers mod 5? Are they quantifiable? Are they numbers?

Just trying to understand what you mean by "quantifiable."

Last edited: May 13, 2012
14. May 13, 2012

### Number Nine

What is the value of aleph null? Methinks a good course in abstract algebra would clear up a lot of these restricted notions of what constitutes a number.

15. May 13, 2012

### uperkurk

PI's value is not infinity but the concept is.

9 followed by ∞ 9's or 1 followed by ∞ 1's?

The 9's have a higher value but they are equal to each because they have ∞ numbers trailing...

16. May 13, 2012

### micromass

None of these are real numbers.

17. May 13, 2012

### uperkurk

Regardless of how people try to justify infinity, the bottom line is if something goes on forever it can never have a number, a value yes but not a number

18. May 13, 2012

### Number Nine

You seem to be treating "number" as synonymous with "real number"; which is to say, "something you can count with". Number systems are just algebraic structures; the elements of those structures are called numbers, whatever they may be. It may disturb you to know that every real number is actually a collection of intervals on the rational number line (a classic construction of the real numbers: Dedekind cuts). We call each of these intervals a real number because it's convenient, and they behave the way we expect real number to behave.

Do you object to the existence of cardinal numbers? Ordinal numbers?

Last edited: May 13, 2012
19. May 13, 2012

### micromass

The thing is that there is no such thing as a "number" in mathematics. We do have

- Rational numbers
- Real numbers
- Surreal numbers
- Transfinite numbers

and many more. Some of these systems allow some form of infinity. But mathematicians never speak of just numbers.

20. May 13, 2012

### Studiot

One reason we have infintity or infinities is that it is sometimes possible and even useful to evaluate expressions of the type

$$\frac{\infty }{\infty }$$

Wallis product is a good example.

21. May 13, 2012

### phoenixthoth

I'm surprised that unlimited hypernatural numbers haven't been mentioned much...

22. May 14, 2012

### coolul007

Don't we use infinity to define a set of numbers, as in a series, to reach a limit or it is limitless. Isn't the underlying premise of Calculus infinite members of a set?

23. May 14, 2012

### HallsofIvy

Yes, but that has nothing to do with the question of whether infinity "is a number or a concept".

24. May 14, 2012

### SteveL27

Well, yes and no. Logically if you build calculus from the ground up, you need a theory of the real numbers, which requires a theory of infinite sets. This is the modern view.

But historically, Newton and Leibniz developed calculus without any set theory and without a rigorous theory of limits. So you don't need infinite sets to do calculus, only to logically justify it.

25. May 15, 2012

### coolul007

As I so UN-eloquently put it, my intent was to show the use of infinity as a concept. i also agree with the previous posts that all numbers are a figment of our imagination, and we have given rules to how they are to behave.