After studying the order of an element, we now return to the standard examples of groups and examine them with greater confidence. The order results become meaningful only after students can recognize familiar groups quickly and correctly. Number systems provide the first reliable family of examples because ordinary addition and multiplication have well-known algebraic properties. In this lesson, students will review groups from number systems, distinguish additive and multiplicative notation, and learn why certain familiar number systems fail to be groups under some operations.

Let $G$ be a non-empty subset of a number system such as $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, or $\mathbb{C}$. If $G$ is a group under ordinary addition or ordinary multiplication, then $(G,+)$ or $(G,\cdot)$ is called a $\textbf{number system group}$.