# Does infinity - infinity equal 0

## Main Question or Discussion Point

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

mathwonk
Homework Helper
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.

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

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

arildno
Homework Helper
Gold Member
Dearly Missed
Anzas said:
then why do we use arithmetic with complex numbers? they aren't really numbers after all
$$\sqrt{-1}$$ is not anymore defined than $$\infty$$
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").

Zurtex
Homework Helper
Anzas said:
then why do we use arithmetic with complex numbers? they aren't really numbers after all
$$\sqrt{-1}$$ is not anymore defined than $$\infty$$
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).

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.

HallsofIvy
Homework Helper
Anzas said:
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.
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"!

Zurtex
Homework Helper
Anzas said:
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.
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?

Zurtex said:
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?
Well I do have -$25 in my account right now! Zurtex Science Advisor Homework Helper whozum said: Well I do have -$25 in my account right now!
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. It was a joke, Stop being critical! :P mathwonk Science Advisor Homework Helper but he could deposit$25 and show the resuilting zero balance!

Hurkyl
Staff Emeritus
Gold Member
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!)

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

Finite Infinity...

Infinity in cosmology
An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point.
antinomies said:
... I can see in space the possibility of any finite experience ... we recognise [the] essential infinity of space in its smallest part." "[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room."

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.
$$\infty = \infty$$
$$\boxed{|\infty| - |\infty| = 0}$$

antinomies said:
... what is infinite about endlessness is only the endlessness itself.
Reference:
http://en.wikipedia.org/wiki/Infinity
http://en.wikipedia.org/wiki/Number

Last edited:
Zurtex
Homework Helper
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:
Alkatran
Homework Helper
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:
$$\infty = \infty - \infty = 0$$

$$\infty = 0$$