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.

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:

where u, v and w are the components (also the notation |x| is used). 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.