# Understanding the large curly brace in a statement about absolute value

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.

• The absolute value of $a$ is $a$ if $a$ is non-negative and $-a$ if $a$ is negative. Mar 27, 2021 at 5:57
• The phrase you're looking for is probably "piecewise function" (also "piecewise defined function" is used). The curly braces are a standard way of giving formulas for them. Mar 27, 2021 at 6:02
• As noted in @Brian's answer, that's the standard way of expressing conditional "cases" in an equation. FYI: The TeX markup for the structure is, fittingly, \begin{cases} ... \end{cases} (which I used in editing your question to transcribe the quote from the image).
– Blue
Mar 27, 2021 at 7:14

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 $$|a|=-a$$.