Complex

Constants

Symbol	Description
ImaginaryUnit	\imaginaryI	The imaginary unit, solution of x^2+1=0

Functions

["Real", z]

\Re(3+4\imaginaryI)

$$\Re(3+4\imaginaryI)$$

Evaluate to the real part of a complex number.

["Real", ["Complex", 3, 4]]
// ➔ 3

["Imaginary", z]

\Im(3+4\imaginaryI)

$$\Im(3+4\imaginaryI)$$

Evaluate to the imaginary part of a complex number. If z is a real number, the imaginary part is zero.

["Imaginary", ["Complex", 3, 4]]
// ➔ 4

["Imaginary", "Pi"]
// ➔ 0

["Conjugate", z]

z^\ast

$$z^\ast$$

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

["Conjugate", ["Complex", 3, 4]]
// ➔ ["Complex", 3, -4]

["Abs", z]

|z|

$$|z|$$

\operatorname{abs}(z)

$$\operatorname{abs}(z)$$

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}

["Abs", ["Complex", 3, 4]]
// ➔ 5

["Arg", z]

\arg(z)

$$\arg(z)$$

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}

["Arg", ["Complex", 3, 4]]
// ➔ 0.9272952180016122

["AbsArg", z]

Return a tuple of the magnitude and argument of a complex number.

This corresponds to the polar representation of a complex number.

["AbsArg", ["Complex", 3, 4]]
// ➔ [5, 0.9272952180016122]

3+4\imaginaryI = 5 (\cos 0.9272 + \imaginaryI \sin 0.9272) = 5
\exponentialE^{0.9272}

["ComplexRoots", z, n]

\operatorname{ComplexRoot}(1, 3)

$$\operatorname{ComplexRoot}(1, 3)$$

Retrurn a list of the n^th roots of a number z.

The complex roots of a number are the solutions of the equation z^n = a.

• Wikipedia: n^th root
• Wikipedia: Root of unity
• Wolfram MathWorld: Root of unity

// The three complex roots of unity (1)
["ComplexRoots", 1, 3]
// ➔ [1, -1/2 + sqrt(3)/2, -1/2 - sqrt(3)/2]