Axiom 2: Associative Addition

The Associative Addition Axiom is one of the properties that every school has taught to their students. The other common rules or laws are the commutative property and the distributive property. These tend to define how we add when using the real numbers, and so it seems natural to use this property even if you don’t fully know or understand you’re using it.

Associative Addition actually can be summed up in this simple statement: a+(b+c)=(a+b)+c. What does this say? This mathematical equation says that you can add numbers together in any order, or that order doesn’t matter. You can add the first two numbers together and then the last one, or you can add the final two together and then the first. In either case, you will get the same answer. Those number systems that follow this property are one step closer to showing that they are indeed a ring. As you will find out, all of these addition axioms can't be casually used to describe subtraction. If you try turning these plus signs into minus signs, you will see that this property won't hold.

The real numbers are an easy example to use when discussing any axioms, for reasons you will find under the Fields page, but since it’s the one we are most familiar with, it’s easy to see why the real numbers follow the Associative property. It has been taught to us since we were young, and thus it doesn’t really require convincing that it exists. In fact, when trying to prove if a number system of some sort is a ring, this is one of the easiest to prove.

As a simple example, let’s say Sarah has 3 baskets of oranges. One basket has 10 oranges, another has 12, and the final has 8. Notice that is doesn’t matter which order we had the oranges up in. In any scenario, we will end up with Sarah having 30 oranges. This is one of the nice things about this property, as it intuitively makes a lot of sense, even to those who aren’t mathematicians. It’s one of those properties that feels like it should work

This quick video provides another quick example of how the Associative Property of Addition works.