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.
The complex conjugate of a given complex number a + bı is a - bı.
The set of all numbers of the form a + bı, where a and b are real
numbers. The real numbers are all complex numbers.
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
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
A number that can be expressed in the form bı, where b is a real
The bı term in every complex number a + bı.
The imaginary unit is ı
. ı =
for a complex number a + bı
. In the complex
plane, it is the distance between the plot of a complex number and the
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.
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.
In a complex number a + bı, a.
For a complex number, a + bı.
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.