Axiom 5: An Additive Inverse

An Additive Inverse is also an axiom that has some precise wording that goes into it. Just like in the Additive Identity axiom, this axiom can be tricky if we only limit our thinking to the real numbers. In the real numbers, we immediately say that we must add to zero to satisfy this axiom. However, it is important to note that this is only true for the real numbers, and isn’t a general statement that can be applied to all number systems.

The precise wording of this axiom is written in the following way: for each a in R, the equation a+x=0 has a solution in R. What is tricky about this definition is that it is predicated by the phrase “for each a in R”. This means that this additive inverse x must exist for every element in R, not just one. This also means that x will change. It isn’t a static value. To determine if this axiom truly holds for a number system, we have to make sure it holds for every single element. If it fails even once, then this axiom doesn’t hold.

For example, let’s use the real numbers as a good example of the additive inverse. In the real numbers, we have the inclusion of negative numbers, which when additive to their respective positive counterpart produce the additive identity, zero. I’ll also reiterate that the emphasis is on producing the additive identity, not zero, even though zero happens to be the additive identity for the real numbers. Since this is true for all numbers in the real number system, the real numbers do follow this axiom.

A number system that doesn’t follow this is actually really easy to determine. Note that in our real number example, it was vital that we were able to use the negative numbers to have an additive inverse. As such, let us look at just the positive real numbers. Hopefully the problem here is really obvious. No 2 positive real numbers add together to get the additive identity, so this system doesn’t have an additive inverse.

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.