Magnitude (mathematics)
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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:
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.
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:
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.