Magnitude
In
science,
magnitude refers to the numerical size of something: see
orders of magnitude.
In mathematics, the magnitude of an object is a nonnegative 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 nearinfrared 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.
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.
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 + 4
i is 5.
Euclidean vectors
The magnitude of a vector of real numbers in a
Euclidean nspace 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 √(4
^{2} + 5
^{2} + 6
^{2}) = √77 or about 8.775.
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: