I'm trying to learn from the book A First Course in Calculus by Serge Lang, but I don't know how to interpret the large curly brace after the absolute value of $a$:
Theorem 2.1. If $a$ is any number, then
$$|a| = \begin{cases}
\phantom{-}a & \text{if}\; a\geq 0 \\
-a & \text{if}\; a < 0 \end{cases} $$
I don't know what the correct name for it either, so I didn't know what to search for.
So, how do I spell this theorem out in sentenced English?
I'm thinking that might help me understand it better.
Best Answer
It’s a definition by cases; here the cases are $(1)$ $a\ge 0$ and $(2)$ $a<0$. Thus, it means that if $a\ge 0$, then $|a|=a$, and if $a<0$, then $|a|=-a$.
This is a very standard way of expressing a definition by cases; you are likely to see it over and over again.