This section in pdf form. complex.pdf
It's shown that generalized complex numbers arise naturally from the law of cosines, , where side is opposite . These numbers can be plotted on an "extended" Argand diagram using the natural coordinates where the imaginary axis is rotated clockwise degrees out of the negative real axis. Similarly, the conjugate axis is rotated counter clockwise degrees out of the negative real axis. Hence if then the imaginary and conjugate axes are coincident resulting in the "standard" Argand diagram, and the "standard" complex numbers.
Consider the triangles in fig (4). From the law of cosines, and . If
(i) | and | (multiplicative and additive closure) | ||
(ii) | and | (commutativity) | ||
(iii) | and | (associativity) | ||
(iv) | (distributive) | |||
(v) | is the additive identity. | |||
(vi) | then is its additive inverse. | |||
(vii) | is the multiplicative identity. | |||
(viii) | and not both then the multiplicative inverse is | |||
The multiplicative inverse was found by noting that if is the multiplicative inverse of then
Hence,
So, from Cramer's rule,
can be written as a sum. Note that and is equivalent to ; so let then can be written as . I want to find an expression for . We have,
Hence
If and then
And,From equation (43), .
The real part of is
and the imaginary part of is
Note: If and then
From equation(43), we know that if is plotted in the plane, then , the distance from the origin to the point , is . This can be accomplished if the positive axis is rotated degrees clockwise out of the axis as in figure (5). Rotate the positive conjugate axis counterclockwise 90 degrees out of the negative real axis. Then can be plotted along with , as shown in figure (5).
In figure (6)
Then
And
In equation (44), for , if and only if . That is, if and only if the imaginary and conjugate axes are rotated degrees from the axis, clockwise and counterclockwise respectively in figure (5). This brings them into coincidence at right angles to the real axis as shown in figure (7).
Replace with throughout. Since , multiplication becomes , , and . Then becomes the field of complex numbers.