• ### Argument

The angle created by the positive real axis and the segment connecting the origin to the plot of a complex number in the complex plane.

• ### Complex Conjugate

The complex conjugate of a given complex number a + bı is a - bı.

• ### Complex Number

The set of all numbers of the form a + bı, where a and b are real numbers. The real numbers are all complex numbers.

• ### Complex Plane

A plane with two perpendicular axes, the real axis and the imaginary axis, on which a complex number a + bı is plotted at the coordinate (a, b). It is customary for the real axis to coincide with the x-axis of the rectangular coordinate system, and for the imaginary axis to coincide with the y-axis of the rectangular coordinate system.

• ### Imaginary Axis

The axis in the complex plane that customarily coincides with the y-axis of the rectangular coordinate system, and on which the imaginary part bı of the complex number a + bı is plotted.

• ### Imaginary Number

A number that can be expressed in the form bı, where b is a real number.

• ### Imaginary Part

The bı term in every complex number a + bı.

• ### Imaginary Unit

The imaginary unit is ı. ı = .

• ### Modulus for a complex number a + bı. In the complex plane, it is the distance between the plot of a complex number and the origin.

• ### Polar Form of a Complex Number

The polar form of a complex number z = a + bı is this: z = r(cos(θ) + ısin(θ)), where r = | z| and θ is the argument of z.

• ### Real Axis

The axis in the complex plane that typically coincides with the x-axis of the rectangular coordinate system, and on which the real part a of a complex numbers a + bı is plotted.

• ### Real Part

In a complex number a + bı, a.

• ### Standard Form

For a complex number, a + bı.

• ### Formulae

 De Moivre's Theorem Let z = r(cos(θ) + ısin(θ).Thenzn = [r(cos(θ) + ısin(θ)]n = rn(cos(nθ) + ısin(nθ), where n is any positive integer.

 Roots of a Complex Number A complex number z = r(cos(θ) + ısin(θ) has exactly nnth roots given by the equation [cos( ) + ısin( )], where n is a positive integer, and k = 0, 1, 2,..., n - 2, n - 1.