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#1 2006-03-30 09:06:45

MathsIsFun
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Complex Number Formulas

Complex Number Formulas


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman
 

#2 2006-04-03 00:27:18

ganesh
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Re: Complex Number Formulas

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:-


Character is who you are when no one is looking.
 

#3 2006-04-04 00:59:38

ganesh
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Re: Complex Number Formulas

If









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


Division of two complex numbers:-


Character is who you are when no one is looking.
 

#4 2006-04-04 01:15:28

ganesh
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Re: Complex Number Formulas

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.


Character is who you are when no one is looking.
 

#5 2006-04-09 15:09:52

ganesh
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Re: Complex Number Formulas

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.


Character is who you are when no one is looking.
 

#6 2006-04-09 15:18:49

ganesh
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Re: Complex Number Formulas

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.


Character is who you are when no one is looking.
 

#7 2006-04-09 16:00:54

ganesh
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Re: Complex Number Formulas

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


Character is who you are when no one is looking.
 

#8 2006-04-09 16:05:08

ganesh
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Re: Complex Number Formulas

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
.


Character is who you are when no one is looking.
 

#9 2009-03-18 18:28:08

ganesh
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Re: Complex Number Formulas

Division and Exponentiation



.


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#10 2009-03-18 18:38:34

ganesh
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Re: Complex Number Formulas

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

.


Character is who you are when no one is looking.
 

#11 2009-03-18 18:49:32

ganesh
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Re: Complex Number Formulas

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

   

   




Character is who you are when no one is looking.
 

#12 2009-03-31 18:10:10

Identity
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Re: Complex Number Formulas

Properties of the magnitude:





  (Triangle Inequality)

 

#13 2011-08-10 19:43:25

Ritu
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Re: Complex Number Formulas

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

Last edited by Ritu (2011-08-10 19:44:37)

 

#14 2011-08-10 19:58:19

bobbym
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Re: Complex Number Formulas

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:


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#15 2011-08-10 20:02:29

anonimnystefy
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Re: Complex Number Formulas

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


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
 

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