Complex Number Formulas

MathsIsFun · 2006-03-29 10:06:45

Jai Ganesh · 2006-04-02 02:27:18

Complex Numbers
A complex number is a number of the form a+bi and it consists of the real part and the imaginary part.
In a+bi, a is the real part and bi is the imaginary part.
i is an imaginary number, i=√(-1)
In polar form the complex number is represented as
r(Cosθ +iSinθ)
rCosθ =a and rSinθ =b; tanθ =b/a.

r is the modulus and θ is the argument.
In exponential form, a complex number is represented as

De Moivre's theorem:-

Jai Ganesh · 2006-04-03 02:59:38

If

For every complex number z, there exists and inverse such that

Division of two complex numbers:-

Jai Ganesh · 2006-04-03 03:15:28

The conjugate of a complex number z=a+bi is given by

Some of the properties of conjugates are

=
=

Re(z) is the Real part of x and Im(z) is the imaginary part of z.

Jai Ganesh · 2006-04-08 17:09:52

Cube roots of unity

Let z denote a cube root of unity.

or

Properties of cube roots of unity:-
(1) Each of the complex roots of cube root of unity is square of the other.
(2) Sum of the cube roots of unity is zero. i.e.

where 1, are the cube roots of unity.

Jai Ganesh · 2006-04-08 17:18:49

Applications of De Moivre's Theorem in finding Roots of Complex Numbers

Let z = x +iy
In polar form,

where
and

where k=0,1,2,3,...(n-1).
This gives n distinct roots of z.

Jai Ganesh · 2006-04-08 18:00:54

nth roots of unity

Let x be a root of unity. Then

where r=0,1,2,....n-1 using De Moivre's Theorem.

Let

The nth roots of unity are where r=0,1,2,3,4...(n-1).
That is, the nth roots of unity are

Jai Ganesh · 2006-04-08 18:05:08

Properties of arguments

(1) The argument of a positive real number is zero.

(2) The argument of a negative real number is

.

(3) The argument of a positive imaginary number is

.

(4) The argument of a negative imaginary number is

or .

Jai Ganesh · 2009-03-17 19:28:08

Division and Exponentiation

.

Jai Ganesh · 2009-03-17 19:38:34

de Moivre's identity for powers of Complex Numbers of Real n

.

Jai Ganesh · 2009-03-17 19:49:32

Powers of Comples Numbers

A power of complex number z to a positive integer exponent n can be written in closed form as

.

The first few are explicitly

Identity · 2009-03-30 19:10:10

Properties of the magnitude:

(Triangle Inequality)

Ritu · 2011-08-09 21:43:25

(-1)^(1/3) is????

Last edited by Ritu (2011-08-09 21:44:37)

bobbym · 2011-08-09 21:58:19

Hi Ritu;

Question should be asked in "Help Me."

The CAS all return

Which their pages claim is the principal value.

You can view it as the solutions to equation

Which has roots:

anonimnystefy · 2011-08-09 22:02:29

hi Ritu the question has already been answered by ganesh in post#5.

Math Is Fun Forum

#1 2006-03-29 10:06:45

Complex Number Formulas

#2 2006-04-02 02:27:18

Re: Complex Number Formulas

#3 2006-04-03 02:59:38

Re: Complex Number Formulas

#4 2006-04-03 03:15:28

Re: Complex Number Formulas

#5 2006-04-08 17:09:52

Re: Complex Number Formulas

#6 2006-04-08 17:18:49

Re: Complex Number Formulas

#7 2006-04-08 18:00:54

Re: Complex Number Formulas

#8 2006-04-08 18:05:08

Re: Complex Number Formulas

#9 2009-03-17 19:28:08

Re: Complex Number Formulas

#10 2009-03-17 19:38:34

Re: Complex Number Formulas

#11 2009-03-17 19:49:32

Re: Complex Number Formulas

#12 2009-03-30 19:10:10

Re: Complex Number Formulas

#13 2011-08-09 21:43:25

Re: Complex Number Formulas

#14 2011-08-09 21:58:19

Re: Complex Number Formulas

#15 2011-08-09 22:02:29

Re: Complex Number Formulas

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