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.
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 -3 + 4i is 5.
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:
- <math>s=\sqrt{u^2+v^2+w^2}</math>
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.
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:
