Physicist: Several questions about doing basic math with infinity have been emailed over the years, so here’s a bunch of them! (More can be added later)
Infinity comes in a lot of shapes and flavors. However, the most straightforward infinity that makes the most intuitive sense (for most people) is probably the infinity that “sits at the end of the number line“. is defined as the “value” such that given any number, x, we always have x < . “Value” is in quotes there (‘ ” ” ‘) because infinity isn’t an actual value, it’s more of a place-holder for “bigger than anything else”.
As soon as the word “infinity” is dropped anywhere on the internet the tone suddenly becomes a little… philosophical. So, just to be specific: > x, for any number x, and is assumed to be the unique thing that has that property (Nothing more or less).
+ 1 = ?
Like you’d expect, + 1 = . This is because > x-1, for any x, and therefore + 1 > x for any x, and therefore (because is defined to be the only thing with this property), +1 = . This brings up the interesting fact that + 1 is not bigger than (no matter what any second grader might say). They’re the same.
+ = ?
again. Pick any number, x, and you find that + > + x/2 > x/2 + x/2 = x. So, + = . This one isn’t too surprising either.
– = ?
This is a bit more nuanced. Your first inclination might be “0”, but keep in mind that that would mean that – = ( + 1) – = 1. The “not-a-number-ness” of infinity means that subtracting it from itself doesn’t make sense, or at the very least, doesn’t have a definitive result.
/ = ?
This is just as nuanced. You may think “1”, because you’re probably more reasonable than not. But consider, if that were the case, then: / = ( + )/ = 2. So, again, there’s no definitive result.
Is even or odd?
Something is even if, when divided by two, the result is an integer. But is /2 an integer? is generally considered to not be an integer (or rational, or even irrational), so isn’t generally considered to be either even or odd.
On a case-by-case basis you can sometimes have “disagreeing infinities” and figure out what they equal. For example, involves a positive infinity () and a negative infinity (), but if you add up the sum one terms at a time you find that it equals ln(2) = 0.6931…
But in general, the operations we freely use with ordinary numbers (addition, subtraction, ) need to be considered very, very carefully before they’re applied to infinities (or even zeros). In fact, mathematicians almost never “plug in “. Instead, they “sneak up” on it using by limits. For example, if you want to figure out what 1/ is, you say “what is the limit as the number x gets arbitrarily large of the function 1/x?”. In this case you can reasonably say that 1/ = 0, but without actually plugging in weird stuff that doesn’t have an actual, numerical, value.