Definition:Absolute Value/Definition 1/Also presented as

Absolute Value: Also presented as

Because $0 = -0$, the value of $\size x$ at $x = 0$ is often included in one of the other two cases, most commonly:

$\size x = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases}$

but this can be argued as being less symmetrically aesthetic.

This form can also be found:

$\size x = \begin{cases} x & : x \ge 0 \\ -x & : x \le 0 \end{cases}$

Sources

• 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.2$: Functions
• 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Variables and Functions
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Exercise $1$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.14$: Modulus
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): absolute value
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): absolute value