# Does infinity - infinity equal 0

1. May 28, 2005

### Orion1

Is $$\infty - \infty = 0$$?

2. May 28, 2005

### mathwonk

not always. for instance, you might notice that the natural, numbers are infinite and also the even numbers are infinite, and then the odd numbers would be infinity minus infinity, but still infinite, not zero, in number. so subtracting infinities is harder than adding or multiplying them.

3. May 29, 2005

### whozum

A mathematician yet not mentioning that arithmetic only applies to numbers! Shame!

4. May 29, 2005

### Anzas

then why do we use arithmetic with complex numbers? they aren't really numbers after all
$$\sqrt{-1}$$ is not anymore defined than $$\infty$$

5. May 29, 2005

### arildno

Both square root of -1 and infinity can be rigorously defined as numbers; neither of them, however will lie on what we call the real number line (or more precisely, they are not "real numbers" in the technical meaning of "real").

6. May 29, 2005

### Zurtex

Complex numbers are really numbers they just are an extention of the real number set and it's fairly easy to prove all the axioms that apply to the real numbers (except inequalities).

7. May 29, 2005

### Anzas

whats the definition of "number" i would say a number is anything that represents some kind of quantity but complex numbers do not they are more like a function than a number.

8. May 29, 2005

### HallsofIvy

Staff Emeritus
Okay, so your "definition" of number has nothing to do with what any mathematician means by "number". Perhaps you should tell us what you mean by "quantity"!

9. May 29, 2005

### Zurtex

Look this up: http://en.wikipedia.org/wiki/Number

The mathematical definition of a number does not have to have a physical representation, ever come across $\pi^e$ apples or -3 bricks?

10. May 29, 2005

Well I do have -$25 in my account right now! 11. May 29, 2005 ### Zurtex No you don't, rather you owe the bank$25 and -$25 is used to represent this difference in direction of who owes who money, you can't actually take out this -$25 and show it to someone.

12. May 29, 2005

### whozum

It was a joke, Stop being critical! :P

13. May 29, 2005

### mathwonk

but he could deposit \$25 and show the resuilting zero balance!

14. May 29, 2005

### Hurkyl

Staff Emeritus
Some mathematicians were having lunch, and saw two people enter a house across the street. Later, they saw three people exit, and concluded that if exactly one person entered the house, it would be empty!

(Okay, this is also supposed to have a physicist and a biologist, but whatever!)

15. May 29, 2005

### Crosson

It sounds like you told the biologist part of the joke.

In general, the joke goes:

Biologist/Engineer: A totally ludicrous generalization of what was just observed.

Physicist: A reasonable conclusion based on what was observed.

Mathematician: An ultra-cautious, over qualified statement which logically follows from the observation.

Perhaps you were telling a math-depreciating joke, and my sense of humor is too old fashion

16. May 29, 2005

### Moo Of Doom

As I heard it:

A physicist, a biologist, and a mathematician were having lunch at a cafe. They watched two people enter the building accross the street. A bit later, they see three people exit. The physicist deduces, "The measurement was inaccurate." The biologist proclaims, "They reproduced." The mathematician then suggests, "Now if one more person enters the building it will be empty."

17. May 29, 2005

### Orion1

Finite Infinity...

Consider the concept of a numerical axis placed upon the surface of a lemniscus Möbius strip, with a philosopher placed at the origin of such an axis. Such a philosopher travelling in either direction would perceive to travel an endless distance into infinity, however, by examining the numerical axis on the Möbius surface, the philosopher perceives the arrival back at the origin.

Therefore, absolute infinity deducted from itself can only equal its origin.
$$\infty = \infty$$
$$\boxed{|\infty| - |\infty| = 0}$$

Reference:
http://en.wikipedia.org/wiki/Infinity
http://en.wikipedia.org/wiki/Number

Last edited: May 29, 2005
18. May 30, 2005

### Zurtex

Orion1 I'm not so sure you are correct, if you consider every element of the real number set and for each element you pair it to another element of the real number set you could make the number of left over elements whatever you liked i.e:

$$\left| \mathbb{R} \right| - \left| \mathbb{R} \right| = \alpha$$

You get set up a correspondence such that the possible solutions for alpha at least exist in the set:

$$\mathbb{Z} \cup \left\{ \aleph_{0}, -\aleph_{0}, \aleph_{1}, -\aleph_{1} \right\}$$

Last edited: May 30, 2005
19. Jun 1, 2005

### Alkatran

Ok. Infinity - infinity is not defined because infinity does not equal 0:

1:
$$\lim_{x \rightarrow \infty} x - \frac{x}{2} = \frac{x}{2}$$

$$\infty - \frac{\infty}{2} = \frac{\infty}{2}$$

$$\infty - \infty = \infty$$

2:
$$\lim_{x \rightarrow \infty} x - x = 0$$

$$\infty - \infty = 0$$

3:
$$\infty = \infty - \infty = 0$$

$$\infty = 0$$