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#1 2006-03-30 09:06:45
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Complex Number Formulas
Complex Number Formulas
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#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
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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
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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
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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
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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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#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
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