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### 17.2 Complex Arithmetic

In the descriptions of the following functions, z is the complex number x + iy, where i is defined as `sqrt (-1)`.

abs (z)

Compute the magnitude of z.

The magnitude is defined as |z| = `sqrt (x^2 + y^2)`.

For example:

```abs (3 + 4i)
⇒ 5
```

arg (z)
angle (z)

Compute the argument, i.e., angle of z.

This is defined as, theta = `atan2 (y, x)`, in radians.

For example:

```arg (3 + 4i)
⇒ 0.92730
```

conj (z)

Return the complex conjugate of z.

The complex conjugate is defined as `conj (z)` = x - iy.

cplxpair (z)
cplxpair (z, tol)
cplxpair (z, tol, dim)

Sort the numbers z into complex conjugate pairs ordered by increasing real part.

The negative imaginary complex numbers are placed first within each pair. All real numbers (those with `abs (imag (z)) / abs (z) < tol`) are placed after the complex pairs.

tol is a weighting factor in the range [0, 1) which determines the tolerance of the matching. The default value is `100 * eps` and the resulting tolerance for a given complex pair is `tol * abs (z(i)))`.

By default the complex pairs are sorted along the first non-singleton dimension of z. If dim is specified, then the complex pairs are sorted along this dimension.

Signal an error if some complex numbers could not be paired. Signal an error if all complex numbers are not exact conjugates (to within tol). Note that there is no defined order for pairs with identical real parts but differing imaginary parts.

```cplxpair (exp (2i*pi*[0:4]'/5)) == exp (2i*pi*[3; 2; 4; 1; 0]/5)
```
imag (z)

Return the imaginary part of z as a real number.