Magnitude (mathematics)

Real numbers

The magnitude of a real number is usually called the absolute value or modulus. It is written | x |, and is defined by:

| x | = x, if x ≥ 0

| x | = -x, if x < 0

This gives the number's distance from zero on the real number line. For example, the modulus of -5 is 5.

Complex numbers

Similarly, the magnitude of a complex number, called the modulus, gives the distance from zero in the Argand diagram. The formula for the modulus is the same as that for Pythagoras' theorem.

$\left|x+iy\right|={\sqrt {x^{2}+y^{2}}}$

For instance, the modulus of -3 + 4i is 5.

Euclidean vectors

The magnitude of a vector x of real numbers in a Euclidean n-space is most often the Euclidean norm, derived from Euclidean distance: the square root of the dot product of the vector with itself:

$\|\mathbf {x} \|:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.$

where x = [x₁, x₂, ..., x_n]. The notation |x| is aslo used for the norm. For instance, the magnitude of [4, 5, 6] is √(4² + 5² + 6²) = √77 or about 8.775.

General vector spaces

A concept of magnitude can be applied to a vector space in general. This is then called a normed vector space. The function that maps objects to their magnitudes is called a norm.

Practical math

A magnitude is never negative. When comparing magnitudes, it is often helpful to use a logarithmic scale. real-world examples include the loudness of a sound (decibel) or the brightness of a star.

To put it another way, often it is not meaningful to simply add and subtract magnitudes.