Changes

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 square of the number's [[Absolute value|magnitude]]

: <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>\tan\theta=\frac{b}{a}</math>,where <math>theta</math> is the phase  

If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand 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.

If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand 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===

===Complex Numbers as Matrices===

The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form

The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form