Absolute Value

📐 Definition

The absolute value returns the nonnegative magnitude of a real input:

|x| = \begin{cases} x, & x \ge 0,\\ -x, & x < 0 \end{cases}

Domain and Codomain

The map $|\cdot|:\mathbb{R}\to[0,\infty)$ is defined for all real inputs and returns nonnegative reals.

⚙️ Key Properties

Even Symmetry

|-x| = |x|

Lipschitz / Triangle Inequality

|x+y| \le |x| + |y|

Derivative (Away from 0)

\frac{d}{dx}|x| = \operatorname{sgn}(x), \qquad x \ne 0

The nondifferentiability at $x=0$ motivates smooth surrogates when gradients are required.

🎯 Special Cases and Limits

$x$	$\lvert x\rvert$
$0$	$0$
$1$	$1$
$-1$	$1$

For $x\ne 0$ , $\lim_{p\to 0^+}|x|^p = 1$ (while $|0|^p=0$ for $p>0$ ), so the limit $p\to 0^+$ produces a discontinuity at the origin.

Log-absolute-value compresses magnitudes in log space. Power functions and fractional powers build magnitude-dependent nonlinearities. Euclidean norms generalize $|x|$ to vectors.

Usage in Oakfield

Oakfield uses absolute value and magnitude computations throughout core operators and diagnostics:

Diagnostics compute mean_abs, max_abs, RMS, and magnitude gates for phase metrics.
Remainder operator supports ABS and LOG_ABS nonlinearities (with fabs/log1p-style handling).
Feature extraction and polar modes rely on magnitudes like $|z|$ to form unit directions $z/|z|$ and to apply thresholding.

Historical Foundations

📜 Magnitude as Distance

Absolute value formalizes “distance from zero” on the real line and is the one-dimensional prototype for norms and metrics.

🌍 Modern Perspective

It remains a core primitive in analysis and numerical modeling, especially for magnitude-based diagnostics and robust objectives.

📚 References

Rudin, Principles of Mathematical Analysis
Kreyszig, Introductory Functional Analysis with Applications