# Complex
## Constants
| Symbol | Description | |
| :-------------- | :------------------ | --------------------------------------------- |
| `ImaginaryUnit` | $$ \imaginaryI $$ | The imaginary unit, solution of $$x^2+1=0$$ |
## Functions
_z_
Evaluate to the real part of a complex number.
```json example
["Real", ["Complex", 3, 4]]
// ➔ 3
```
_z_
Evaluate to the imaginary part of a complex number. If _z_ is a real number, the
imaginary part is zero.
```json example
["Imaginary", ["Complex", 3, 4]]
// ➔ 4
["Imaginary", "Pi"]
// ➔ 0
```
_z_
Evaluate to the complex conjugate of a complex number. The conjugates of complex
numbers give the mirror image of the complex number about the real axis.
$$
z^\ast = \Re z - \imaginaryI \Im z
$$
```json example
["Conjugate", ["Complex", 3, 4]]
// ➔ ["Complex", 3, -4]
```
_z_
Evaluate to the magnitude of a complex number.
The **magnitude** of a complex number is the distance from the origin to the
point representing the complex number in the complex plane.
$$
|z| = \sqrt{(\Im z)^2 + (\Re z)^2}
$$
```json example
["Abs", ["Complex", 3, 4]]
// ➔ 5
```
_z_
Evaluate to the argument of a complex number.
The **argument** of a complex number is the angle between the positive real axis
and the line joining the origin to the point representing the complex number in
the complex plane.
$$
\arg z = \tan^{-1} \frac{\Im z}{\Re z}
$$
```json example
["Arg", ["Complex", 3, 4]]
// ➔ 0.9272952180016122
```
_z_
Return a tuple of the magnitude and argument of a complex number.
This corresponds to the polar representation of a complex number.
```json example
["AbsArg", ["Complex", 3, 4]]
// ➔ [5, 0.9272952180016122]
```
$$
3+4\imaginaryI = 5 (\cos 0.9272 + \imaginaryI \sin 0.9272) = 5
\exponentialE^{0.9272}
$$
_z_, _n_
Return a list of the nth roots of a number _z_.
The complex roots of a number are the solutions of the equation $$z^n = a$$.
- Wikipedia: [nth root](https://en.wikipedia.org/wiki/Nth_root)
- Wikipedia: [Root of unity](https://en.wikipedia.org/wiki/Root_of_unity)
- Wolfram MathWorld: [Root of unity](http://mathworld.wolfram.com/nthRoot.html)
```json example
// The three complex roots of unity (1)
["ComplexRoots", 1, 3]
// ➔ [1, -1/2 + sqrt(3)/2, -1/2 - sqrt(3)/2]
```