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.

  • $\begingroup$ The absolute value of $a$ is $a$ if $a$ is non-negative and $-a$ if $a$ is negative. $\endgroup$ Commented Mar 27, 2021 at 5:57
  • $\begingroup$ 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. $\endgroup$ Commented Mar 27, 2021 at 6:02
  • $\begingroup$ 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). $\endgroup$
    – Blue
    Commented Mar 27, 2021 at 7:14

1 Answer 1


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.


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