1.4 Introduction and Key Insights

The inverse property of addition states that every real number has a special companion which we call its additive inverse. We define this additive inverse in relation to our additive identity, zero. Basically, the sum of a number and its additive equals zero or the additive inverse of a number is its negative. The fact that $5+(-5)=0$ tells us that the number $-5$ is the additive inverse of the number $5$.

Note that there are no additive inverses if we only use positive numbers (that is, the natural numbers)! We must use the larger, richer collection of integers to use our new property.

You can think about this inverse property as "mathematical canceling", where we regard our additive identity, zero, as a "do nothing" or neutral number. The existence of additive inverses simply tells us we can undo, neutralize or cancel every number. We can undo or cancel $10$ by adding $-10$; we can cancel $-42$ by adding $-(-42)=42$. Here is a fun question to ponder: What is the additive inverse of $0$?