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 |

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

## 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*| = √ (*x*² +*y*² )

`i`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:

*u*,

*v*and

*w*are the components. For instance, the magnitude of [4, 5, 6] is √(4

^{2}+ 5

^{2}+ 6

^{2}) = √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.See also: