# What is the result of $\infty - 1$?

I was wondering after reading "What is the result of infinity minus infinity", is there any logical result for $\infty - 1$ ?

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The logical result is $\infty$. Take for example the sequence $n$, then $\lim n=\infty$. Now $\lim (n-1)=\infty$, too, hence $\infty-1=\infty$. This is one possible explanation. – mpiktas Sep 27 '11 at 11:07
Without context it's technically ill-defined, but I think it'll be $\infty$ no matter which way you slice it. – anon Sep 27 '11 at 11:13
See en.wikipedia.org/wiki/… (and could someone please teach me how to embed links better?!?) – The Chaz 2.0 Sep 27 '11 at 12:44
@The Chaz: Use Markdown: write [link to Wikipedia](http://en.wikipedia.org) to get a link to Wikipedia. Right under the "add comment" button there's a help link containing the very basics of Markdown and a link here for further info. – commenter Sep 27 '11 at 13:10
Thanks, @com ! I'm on the mobile site, but will inform myself next time there's a computer available :) – The Chaz 2.0 Sep 27 '11 at 14:47

Most usually, the answer will be "it doesn't make sense to write that down". The operation of subtraction is something defined only for certain classes of numbers. The class you are probably most familiar with are the Real numbers (denoted by $\mathbb{R}$), which you can think of intuitively as numbers that you can point to on a number line. These include things like $2.5$, $\pi$, and $\sqrt{2}$, but it does not include $\infty$. Since subtraction is defined for these numbers only, to then ask "What is $\infty - 1$ ?" is like asking "What is Cat - 1 ?".

In more advanced mathematics (for example, in Measure theory) we do allow $\infty$ to denote a certain object that can interact with the Real numbers. In those situations, it usually follows your intuition and $\infty - 1$ is defined to be $\infty$ again. Another example of when mathematicians consider the idea of "$\infty - 1$" is in introductory calculus, when one first learns about limits. In that case however, the usage of the symbol $\infty$ is purely short hand notational- it doesn't actually denote any type of number. If a function $f$ becomes arbitrarily large as $n$ becomes large, we may write $\displaystyle\lim_{n\to\infty} f(n) = \infty$ and then for shorthand a teacher may write on the board $$\displaystyle\lim_{n\to\infty} f(n)-1 = \infty - 1 = \infty$$ but strictly that equation is invalid. So indeed, the idea does come up in valid forms, but what you should take away from this post is mainly contained in the first paragraph.

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Ah, "Cat - 1" explained it. – genesis Sep 27 '11 at 11:17
+1, for "Cat - 1." – rcollyer Sep 27 '11 at 15:23
For me personally, Cat - 1 is often well defined. I invariably used a Cat symbol to represent a constant (instead of, say, c) on physics and math tests in college. – Eric J. Apr 2 '12 at 6:00

In the real number system there is no item "$\infty$". Nor is there in the complex number system. There are some other number systems that DO have such an item. One is called the "Riemann sphere" ... consisting of the complex numbers with an extra point $\infty$. Legitimate caluclations defined on the Riemann sphere do, indeed, include the equation $\infty - 1 = \infty$.

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In layman's terms, infinity is something which you cannot count. Take out $1$ from it and you still cannot count it, so the result is infinity.

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This answer just adds to the confusion of the question author. See the answer by @RagibZaman to see a good answer. – 5xum Jul 10 '14 at 10:44