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→Complex numbers 4s m4trices
===Polar notation===
===Polar notation===
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
: <math>\rho^2=a^2+b^2</math> is the magnitude squared
: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]]
: <math>\theta=\arctan\frac{b}{a}</math> is the phase
: <math>\theta=\arctan\frac{b}{a}</math> is the phase
If a line is drawn on the Argand diagram from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the Argand diagram.
If a line is drawn on the [[complex plane]] from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the complex plane.
===Complex Numbers as Matrices===
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
:[a b]
:[-b a],
with <math>a+bi</math> mapping as a function f to the above matrix.
We can see that F and F' are isomorphic because:
The function f is clearly 1-to-1 and onto,
<math>f(x+y)=f(x)+f(y)</math>,
and <math>f(x*y)=f(x)*f(y)</math>.
===In popular culture===
===In popular culture===