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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Complex Number Formulas

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

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**ganesh****Moderator**- Registered: 2005-06-28
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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.

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**ganesh****Moderator**- Registered: 2005-06-28
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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.

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**ganesh****Moderator**- Registered: 2005-06-28
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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.

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**ganesh****Moderator**- Registered: 2005-06-28
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**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.

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**ganesh****Moderator**- Registered: 2005-06-28
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**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.

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**ganesh****Moderator**- Registered: 2005-06-28
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**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

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**ganesh****Moderator**- Registered: 2005-06-28
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**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.

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**ganesh****Moderator**- Registered: 2005-06-28
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**Division and Exponentiation**

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**ganesh****Moderator**- Registered: 2005-06-28
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**de Moivre's identity for powers of Complex Numbers of Real n**

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**ganesh****Moderator**- Registered: 2005-06-28
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**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

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**Identity****Member**- Registered: 2007-04-18
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**Properties of the magnitude:**

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**Ritu****Member**- Registered: 2011-08-09
- Posts: 1

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

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,677

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 agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,658

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

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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