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# 8E. Complex Numbers

`m0=: cnj=: +` | Conjugate
`m1=: mag=: |` | Magnitude
`m2=: %:@(cnj*])` | "
`m3=: rai=: +.` | Real and imaginary parts
`m4=: maa=: *.` | Magnitude and angle
`m5=: irai=: rai^:_1` | Inverse `rai`
`m6=: imaa=: maa^:_1` | Inverse `maa`
`m7=: rou=: [:^ 0j2p1&% * i.` | Roots of unity
`m8=: rpg=: rai@rou` | Regular polygon
`d9=: zero=: ] * 10&^@-@[ < |` | Zero any real y less than 10^-x in mag
`m10=: z=:({.,{:*1e_6"_<%~/@:|)&.rai` | Zero imaginary if relatively small
`m11=: (1e_10&$:) : (j./"1@((] * (<:|)) +.))` | Clean y

The function `z` may be used to zero any imaginary part that is relatively small compared to the corresponding real part. For example:

       (] ,: z) a=:3+j.10^-2*i. 5
    3j1 3j0.01 3j0.0001 3j1e_6 3j1e_8
    3j1 3j0.01 3j0.0001      3      3

       z a
    3j1 3j0.01 3j0.0001 3 3

Complex numbers can be scaled by multiplication by a real number, and shifted and rotated by addition and multiplication by complex numbers. For example:

       ]a=: rou 3 Third roots of unity
    1 _0.5j0.866025 _0.5j_0.866025

       |a Lie on the unit circle
    1 1 1

       1ad30 Complex of mag 1 and angle of 30 degrees
    0.866025j0.5

       6&zero@rai&.> (] ; 3j4&+ ; 1ad30&* ; 2ad60&*) a
    +---------------------------------------------------+
    |   1        0|  4       4| 0.866025 0.5| 1  1.73205|
    | 0.5 0.866025|2.5 4.86603|_0.866025 0.5|_2        0|
    |_0.5 0.866025|2.5 3.13397|        0  _1| 1 _1.73205|
    +---------------------------------------------------+
     Coordinates  Shift by 3,4   Rotate by   Rotate by
     of triangle                 30 degrees   60 degrees

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