complex number questions (1 Viewer)

[ordnaiv]

hey guys, i just have a couple of questions to ask:

1. prove by mathemtical induction that [z^n] = [z]^n

and arg(z^n) = n arg z

2. use the properties of modulus and argument of a complex number to deduce that the conjugate of the products of z1 and z2 = conjugate of z1 x the conjugate of z2

thanks.

 

[pLuvia]

1.
Show true for n=1
Assume n=k
Consider n=k+1
[z^k+1]
=[z^k.z]
From assumption
[z^k]=[z]^k
=[z]^k.[z]
=[z]^k+1

Hence by mathematical induction etc etc

2.
Let z₁=rcis@
Let z₂=kcis#

Then find the conjugates and you will be able to work it out

Eg. z₁.z₂=rcis@.kcis#

 

[ordnaiv]

Oh I See, thank you for all the help :

 

[ordnaiv]

hey how about for the mathematical induction of the arg question?

 

[Riviet]

Assume arg(z^k)=k.arg(z)

Consider n=k+1:

arg(z^k+1)) = arg(z^k) + arg(z) [by asssumption]

=k.arg(z) + arg(z)

=(k+1).arg(z) #

edit: added "by assumption", this is very important.