Complex Number Modulus

A Complex Number Modulus is a norm of the complex number in the complex plane.

• AKA: Complex Norm, Absolute Value.
• Context:
  • It can be denoted [math]\displaystyle{ |z| }[/math] for [math]\displaystyle{ z=a+ib }[/math].
  • It can be defined by [math]\displaystyle{ |z|=\sqrt{a^2+b^2} }[/math] for [math]\displaystyle{ z=a+ib }[/math].
  • If [math]\displaystyle{ z=a+ib }[/math] then |z| can be defined as the distance from the origin [math]\displaystyle{ (0,0) }[/math] to [math]\displaystyle{ (a,b) }[/math] point in the complex plane.
  • …
• Example(s):
  • [math]\displaystyle{ |1+i|= \sqrt{2} }[/math]
  • [math]\displaystyle{ |i|= 1 }[/math]
  • [math]\displaystyle{ |-i|= 1 }[/math]
• Counter-Example(s):
  • Complex Exponentiation Operation.
  • Complex Number Argument.
• See: Complex Number Argument, Complex Exponential Function.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Absolute_value#Complex_numbers 2015-11-22.
  • QUOTE: For any complex number [math]\displaystyle{ z = x + iy, }[/math] where [math]\displaystyle{ x\textrm{ and }y }[/math] are real numbers, the absolute value or modulus of [math]\displaystyle{ z }[/math] is denoted [math]\displaystyle{ |z| }[/math] and is given by

[math]\displaystyle{ |z| = \sqrt{x^2 + y^2}. }[/math]

When the imaginary part [math]\displaystyle{ y }[/math] is zero this is the same as the absolute value of the real number [math]\displaystyle{ x }[/math]

When a complex number [math]\displaystyle{ z }[/math] is expressed in polar form as [math]\displaystyle{ z = r e^{i \theta} }[/math] with [math]\displaystyle{ r \geq 0 }[/math] and [math]\displaystyle{ \theta }[/math] real, its absolute value is [math]\displaystyle{ |z| = r }[/math].

• (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.

1999

• (Mathworld Wolfram, 1999) ⇒http://mathworld.wolfram.com/ComplexModulus.html
  • QUOTE: The modulus of a complex number [math]\displaystyle{ z }[/math], also called the complex norm, is denoted [math]\displaystyle{ |z| }[/math] and defined by

[math]\displaystyle{ |x+iy|=\sqrt{x^2+y^2} }[/math]

If [math]\displaystyle{ z }[/math] is expressed as a complex exponential (i.e., a phasor), then

[math]\displaystyle{ |re^{i\phi}|=|r| }[/math]

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