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

Question

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

Answer 1

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.

Answer 2

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 $\infty - 1 = \infty$.