Infinity minus Infinity

  • Thread starter Holocene
  • Start date
  • #1
231
0

Main Question or Discussion Point

Does subtracting infinity from infinity leave you with zero?

Or could you subtract infinity from infinity and still have infinity?
 

Answers and Replies

  • #2
1,425
1
Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.
 
  • #3
JasonRox
Homework Helper
Gold Member
2,314
3
Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.
You can't subtract but you can add infinity from infinity.

Let N be the natural numbers and R be the real numbers. And card(X) denote the cardinality of X.

card(R) + card(N) = card(R)
card(N) + card(N) = card(N)
 
  • #4
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
Does subtracting infinity from infinity leave you with zero?

Or could you subtract infinity from infinity and still have infinity?
What "infinity" are you talking about? What notion of subtraction are you talking about? Your question really cannot be answered unless these are specified. (Though we can guess at what you meant, in order to give an answer)

Some examples:
The extended real line contains two points "at infinity": [itex]+\infty[/itex] and [itex]-\infty[/itex]. The extended real line also comes with a notion of subtraction that's defined for most, but not all arguments. [itex](+\infty) - (-\infty) = +\infty[/itex] and [itex](-\infty) - (+\infty) = -\infty[/itex], but [itex](+\infty) - (+\infty)[/itex] and [itex](-\infty) - (-\infty)[/itex] are undefined. (And any combination involving at least one finite number is defined)

In the cardinal numbers, there are lots of infinite cardinals. (literally, too many to count) Subtraction makes little sense for them, because if [itex]\alpha \leq \beta[/itex] and [itex]\beta[/itex] is infinite, then [itex]\alpha + \beta = \beta[/itex].

The hyperreal line contains many infinite and infinitessimal numbers, and in a certain sense, the hyperreals behave exactly like the reals. (e.g. you can subtract any hyperreal from any other hyperreal)
 
Last edited:
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,833
956
You can't subtract but you can add infinity from infinity.

Let N be the natural numbers and R be the real numbers. And card(X) denote the cardinality of X.

card(R) + card(N) = card(R)
card(N) + card(N) = card(N)
But, in order to do that, you have to be talking about "cardinal numbers", not the regular real numbers- which I'm pretty sure is what the OP was talking about. "Infinity", in any sense, is not a real number and so neither addition nor subtraction (nor, for that matter multiplication or division) is defined for "infinity".
 

Related Threads on Infinity minus Infinity

  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
4
Views
9K
Replies
11
Views
4K
  • Last Post
Replies
7
Views
9K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
20
Views
3K
Replies
5
Views
1K
Replies
1
Views
2K
Replies
23
Views
4K
  • Last Post
Replies
4
Views
3K
Top