Magnitude

In science, magnitude refers to the numerical size of something: see orders of magnitude.

In mathematics, the magnitude of an object is a non-negative real number, which in simple terms is its length.

In astronomy, magnitude refers to the logarithmic measure of the brightness of an object, measured in a specific wavelength or passband, usually in optical or near-infrared wavelengths: see apparent magnitude and absolute magnitude.

In geology, the magnitude is a logarithmic measure of the energy released during an earthquake. See Richter scale.

Table of contents

1 Real numbers
2 Complex numbers
3 Euclidean vectors
4 General vector spaces

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". 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.

| x + iy | = √ ( x² + y² )

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

Euclidean vectors

The magnitude of a vector 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. For instance, the magnitude of [4, 5, 6] is √(4² + 5² + 6²) = √77 or about 8.775.

General vector spaces

A concept of length 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.