About

The absolute value represents the distance of a number or vector from the origin (zero point) regardless of direction. While trivial for integers the concept expands into the "Modulus" for complex numbers and the "Norm" for vectors in linear algebra. This tool is designed for students and engineers needing precise calculations for |x|.

It handles multiple domains. For real numbers it returns the non-negative value. For complex numbers a + bi it calculates the magnitude using the Pythagorean theorem. For vectors it computes the Euclidean norm (L2 norm). Additionally it includes an algebraic solver for equations like |ax + b| = c splitting the problem into its positive and negative cases to find all valid solutions.

Formulas

The calculation depends strictly on the input domain.

Complex Modulus: For z = 3 − 4i

|z| = √3² + (−4)² = 5

For algebraic equations we must isolate the absolute value term and then bifurcate the equation into two linear equations to solve for the unknown variable.

Reference Data

Type	Notation	Formula
Real Number	\|x\|	{ x if x ≥ 0−x if x < 0
Complex Number	\|z\|	√a² + b²
Vector (2D)	\|v\|	√x² + y²
Vector (3D)	\|v\|	√x² + y² + z²
Equation	\|x\| = a	x = a ∨ x = −a

Frequently Asked Questions

It is the positive version of that number. For example |-5| is 5. Geometrically it represents the distance from zero on the number line which cannot be negative.

Use the Pythagorean theorem. Square the real part, square the imaginary part, add them together, and take the square root of the result.

Yes. An equation like |x| = -3 has no solution because an absolute value can never be negative (in the real number system).

The Euclidean Norm is simply the length of a vector. It is the square root of the dot product of the vector with itself.