What is ∞ -(∞-1)?

Question

I have come up with two possible answers, but am unsure which is true. The first is that ∞-1 is still ∞,as ∞ is endless, and if taking away 1 made it not endless, then ∞ would not be endless either, it would just be one more then the number you got by subtracting 1 from ∞. Then, when you take ∞ away from ∞, you would get 0, as x-x=0, no matter the value of x. Therefore, ∞-(∞-1) would equal 0. My other possible answer is that x-(x-1) should equal one, because that x-1 equals 1 less than x, and a number minus one less than itself equals one more then zero. Therefore, ∞-(∞-1) would equal 1. I am not sure whether to base the answer off the fact that x-(x-1)=1, or that ∞-1=∞.

it would be more appropriate to write $\lim_{x \to \infty} x-(x-1)$ — Albus Dumbledore, Oct 09 '20 at 17:51
This is undefined. Subtracting infinities is not a well-defined operation. — Cameron Williams, Oct 09 '20 at 17:52
Well, we could add symbols $\infty, -\infty, \infty-\infty, -\infty+\infty, ...$ to $\mathbb{R}$ and obtain some sort of algebraic structure — Jakobian, Oct 09 '20 at 17:55
@AndreasBlass I think that the sequence of symbols could be given various meanings - the point is that mathematical symbols derive meaning (or properties at least) from their definition, and no definition is here provided ... — Mark Bennet, Oct 09 '20 at 18:26
This is addressed well in this answer by Eevee Trainer to "Additive inverse of infinity". — Mark S., Oct 09 '20 at 19:33

score 2 · Answer 1 · answered Oct 09 '20 at 17:51

2

$\infty-(\infty-1)$ has the same status as $\infty-\infty$: an indeterminate form. Neither $\infty$ or $1$ is its resolution all the time, and the actual answer depends on where you got the form from.

answered Oct 09 '20 at 17:51

Parcly Taxel

103,344

I wrote another answer below to include an example :) – Riemann'sPointyNose Oct 09 '20 at 18:22

score 1 · Answer 2 · answered Oct 09 '20 at 18:22

It's still undefined. You would agree that

$${\lim_{x\rightarrow \infty}x=\infty}$$

and also that

$${\lim_{x\rightarrow \infty}(x-1)=\infty}$$

Now, if you take this limit:

$${\lim_{x\rightarrow \infty}x-(x-1)}$$

the answer, just by Algebraic simplification, is ${1}$. And the expression is approaching the form ${\infty-(\infty-1)}$. Equally, you can see that

$${\lim_{x\rightarrow \infty}x-((x-1)-1)}$$

is also approaching the form ${\infty-(\infty-1)}$ (since as we said, ${\lim_{x\rightarrow \infty}(x-1)}$ still approaches infinity). Yet the limit (again, through Algebraic simplification) is ${2}$. We have approached the expression in two different ways and got two different answers.

What is ∞ -(∞-1)?

2 Answers2