Zero = infinity?

7 Sep 2021

In some contexts, zero and infinity behave identically.

Define S to be the set of nonnegative real numbers: [0, ∞). Within this set, multiplication has the following properties:

• Closure: ab ∈ S
• Commutativity: ab = ba
• Associatitivy: (ab)c = a(bc)
• Annihilation: 0a = 0
• Identity: 1a = a

where a, b, and c can be any elements of the set S. In addition, every value except for zero has an inverse. More precisely, ∀a ∈ S, a ≠ 0. ∃!b ∈ S. ab = 1.

What happens if we replace 0 with infinity? It turns out that all of the above properties still hold!

Define S to be the set of positive real numbers along with infinity: (0, ∞]. Within this set, multiplication has the following properties:

• Closure: ab ∈ S
• Commutativity: ab = ba
• Associatitivy: (ab)c = a(bc)
• Annihilation: ∞a = ∞
• Identity: 1a = a

where a, b, and c can be any elements of the set S. In addition, every value except for infinity has an inverse. More precisely, ∀a ∈ S, a ≠ ∞. ∃!b ∈ S. ab = 1. We can’t define the inverse of infinity to be zero, because zero is no longer inside of the set S.

This means that adding zero to the set of positive real numbers, and adding infinity to that same set, have the same exact effect on the properties of multiplication. In other words, infinity and zero are indistinguishable if all you’re doing is multiplying things. This similarity surprised me when I first discovered it, because the concepts of “zero” and “infinity” are total opposites of each other.