Axiom 12: Multiplicative Inverse

This is the final axiom. The 12th axiom: a Multiplicative Inverse. This axiom, on top of the other 11 axioms, is what will define a field, which makes this extremely important. Thankfully, unlike the previous Integral Domain, we have all heard of a multiplicative inverse before, so this won’t take nearly as much explaining.

However, just to get on the same page, what is a multiplicative inverse? In layman’s terms, a multiplicative inverse is a number that, when multiplied by another numbers, gets the value “1”. Once again, I will use “1” to refer to the multiplicative identity and “0” to refer to the additive identity. In precise language: For each a not equal to “0” in R, the equation ax=”1” has a solution in R. If we look to the real numbers as a simple example (which actually finally makes sense since this is the final axiom to create a field), every number outside of 0 has a multiplicative inverse. 2x=1. What is x? Well, we know that x=1/2. This means that ½ and 2 are multiplicative inverses of each other.

And that’s it for the 12 axioms! This one was pretty short, especially in comparison to the others, but it isn’t too complicated to understand what a multiplicative inverse is. Continue on to look at two possible extensions to these outcomes: why is a field important and taking a new look at the Complex Numbers!


This video discusses not only this axiom, but also some other axioms. Do note that the video only uses the real numbers as an example, so if your not using real numbers, this might change. Abstractly though, this is the same information as above.