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 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.
- <math> \left| x + iy \right| = \sqrt{x^2 + y^2 }<math>
For instance, the modulus of -3 + 4i is 5.
Euclidean vectors
The magnitude of a vector x 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>||x||=\sqrt{u^2+v^2+w^2}<math>
where u, v and w are the components (also the notation |x| is used).
For instance, the magnitude of [4, 5, 6] is √(42 + 52 + 62) = √77 or about 8.775.
General vector spaces
A concept of magnitude 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.
Practical Math
A magnitude is never negative. When comparing magnitudes, it is often helpful to use a logarithmic scale. real-world examples include the loudness of a sound (decibel) or the brightness of a star.
To put it another way, often it is not meaningful to simply add and subtract magnitudes.
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