Math Is Fun Forum

teo

Member

Registered: 2007-08-04

Posts: 7

complex numbers

Using De Moivre's theorem, express (1-i)^-7 in x+yi form

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to take the first step is easy, to climb up the mountain is another matter.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: complex numbers

Well, I've learned something today, I've never heard of De Moivre's theorem but a few minutes over at wikipedia gave me a clue as to what it is, but I'm having a little trouble applying it. Instead, let's do it the old fashion way:

In step one, I use an exponent rule:

Then I uber-FOILed (made up word there) using pascal's triangle and some basic imaginary rules (actually, I had google do the work, but I can do it here:

I then rationalized it by multiplying the numerator and denominator by the conjugate of 8 + 8i (which is 8-i).

Well, that was fun and time consuming, but it still doesn't really give you what you were looking for.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: complex numbers

The Complex Number Calculator gets 0.0625-0.0625i, so it agrees with you!

But still no De Moivre

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

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: complex numbers

Isn’t it obvious that

Tsk tsk, all of you!

mamoo

Member

Registered: 2007-09-06

Posts: 3

Re: complex numbers

where     

    and 

and by De Moivre Theorem

now consider 1 -i =

since cos(-x) = cos x 

now with De Moivre Theorem
(1-i)^7 =

           =

       
                                   
           =

since cos(-x) = cos x

           =

since sin(7pi/4) = -sqrt(2)/2

           =

           =

           =

since (1/sqrt(2)^6) = (1/8)

           =

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: complex numbers

QED!

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

mamoo

Member

Registered: 2007-09-06

Posts: 3

Re: complex numbers

wat does QED means?

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: complex numbers

Quod erat demonstrandum, a latin term meaning "What was required to be proven has been proved".

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: complex numbers

While the interpretation is correct, I prefer the literal translation:

quod: Which
erat: was to be
demonstrandum: demonstrated

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."