Mathematical Norm

AKA: Complex Number Modulus.
Context:
- It can be denoted as [math]\displaystyle{ ||u|| }[/math] or as [math]\displaystyle{ |u| }[/math]
- It can be defined as the distance between origin [math]\displaystyle{ (0,0) }[/math] to the vector [math]\displaystyle{ u = (a,b) }[/math] point using the Pythagorean Theorem, [math]\displaystyle{ ||u||=\sqrt{a^2+b^2} }[/math]
- It can be defined as the Complex Number Modulus. For any [math]\displaystyle{ z=a+ib }[/math] then [math]\displaystyle{ |z|=\sqrt{a^2+b^2} }[/math]
Example(s):
- [math]\displaystyle{ ||(3,4)||= \sqrt{3^2+4^2}=5 }[/math]
- [math]\displaystyle{ |3+i4|= \sqrt{3^2+4^2}=5 }[/math]
Counter-Example(s):
See: Vector Space, Complex Number Modulus Pythagorean Theorem, Function Normalization.

References

(Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm_of_a_complex_number Retrieved:2015-11-21.
- QUOTE: The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane R². This identification of the complex number [math]\displaystyle{ x + iy }[/math] as a vector in the Euclidean plane, makes the quantity [math]\displaystyle{ \sqrt{x^2 +y^2} }[/math] (as first suggested by Euler) the Euclidean norm associated with the complex number.

(Mathworld Wolfram, 1999) ⇒ http://mathworld.wolfram.com/Norm.html Retrieved:2015-11-19
- QUOTE: The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Norms exist for complex numbers (the complex modulus, sometimes also called the complex norm or simply "the norm"), Gaussian integers (the same as the complex modulus, but sometimes unfortunately instead defined to be the absolute square), quaternions (quaternion norm), vectors (vector norms), and matrices (matrix norms). A generalization of the absolute value known as the p-adic norm is also defined. (...) The term "norm" is often used without additional qualification to refer to a particular type of norm (such as a matrix norm or vector norm). Most commonly, the unqualified term "norm" refers to the flavor of vector norm technically known as the L2-norm. This norm is variously denoted [math]\displaystyle{ ||x||_2, ||x||,\textrm{ or } |x|, }[/math] and gives the length of an n-vector [math]\displaystyle{ x=(x_1,x_2,...,x_n) }[/math]. It can be computed as

[math]\displaystyle{ |x|=\sqrt{x_1^2+x_2^2+...+x_n^2} }[/math]