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. $\endgroup$ 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. $\endgroup$ 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
    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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.