After learning how to verify a group, we now apply the method to the most familiar class of examples: additive groups of number systems. These examples are important because they provide the model for many later algebraic structures. In an additive group, the identity element is usually $0$, and the inverse of an element $a$ is usually $-a$. The main work is to check that the set is closed under addition and that additive inverses remain inside the set. In this lesson, students will verify additive groups such as $(\mathbb{Z},+)$, $(m\mathbb{Z},+)$, $(\mathbb{Q},+)$, $(\mathbb{R},+)$, and $(\mathbb{C},+)$.

Let $G$ be a non-empty set with addition $+$. Then $(G,+)$ is called an $\textbf{additive group}$ if $(G,+)$ is a group. In additive notation, the identity element is denoted by $0$, and the inverse of $a\in G$ is denoted by $-a$.