# 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                        NB. Third roots of unity
1 _0.5j0.866025 _0.5j_0.866025

|a                                NB. Lie on the unit circle
1 1 1

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