: Mathématiques: modulus of a complex number

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modulus of a complex number

Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number ${\\displaystyle z=x+iy,}$ where ${\\displaystyle x}$ and ${\\displaystyle y}$ are real numbers, the absolute value or modulus of ${\\displaystyle z}$ is denoted ${\\displaystyle |z|}$ and is defined by ${\\displaystyle |z|={\\sqrt {\\operatorname {Re} (z)^{2}+\\operatorname {Im} (z)^{2}}}={\\sqrt {x^{2}+y^{2}}},}$ the Pythagorean addition of ${\\displaystyle x}$ and ${\\displaystyle y}$ , where ${\\displaystyle \\operatorname {Re} (z)=x}$ and ${\\displaystyle \\operatorname {Im} (z)=y}$ denote the real and imaginary parts of ${\\displaystyle z}$ , respectively. When the imaginary part ${\\displaystyle y}$ is zero, this coincides with the definition of the absolute value of the real number ${\\displaystyle x}$ . When a complex number ${\\displaystyle z}$ is expressed in its polar form as ${\\displaystyle z=re^{i\ heta },}$ its absolute value is ${\\displaystyle |z|=r.}$ Since the product of any complex number ${\\displaystyle z}$ and its complex conjugate ${\\displaystyle {\ar {z}}=x-iy}$ , with the same absolute value, is always the non-negative real number ${\\displaystyle \\left(x^{2}+y^{2}\ ight)}$ , the absolute value of a complex number ${\\displaystyle z}$ is the square root of ${\\displaystyle z\\cdot {\\overline {z}},}$ which is therefore called the absolute square or squared modulus of ${\\displaystyle z}$ : ${\\displaystyle |z|={\\sqrt {z\\cdot {\\overline {z}}}}.}$ This generalizes the alternative definition for reals: ${\ extstyle |x|={\\sqrt {x\\cdot x}}}$ . The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity ${\\displaystyle |z|^{2}=|z^{2}|}$ is a special case of multiplicativity that is often useful by itself.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Absolute_value#Complex_numbers)