SN PLUS 3

SN PLUS 3

Complex Numbers

Study Guide

Terms and Formulae

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Terms and Formulae

Terms

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.

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