# Thread: Consider the complex number

1. ## Consider the complex number

a)Consider the complex number z1=-3+4i

i) Express this number as z1=re^iθ

ii) Find ln(z1)

SO WE KNOW WHAT r=5

AND WE KNOW THAT θ=126,... ?

here i have a problem - i dont know how to slove the a angle (this angle must be with PI) having the angle i know how to but to z1=re^iθ

I dont know how to solve ii) i found on wikipedia the rule but i dont understnd - can somebody help ?

2. Er you just convert the degree to radian.

We know that 180 degrees correspond to pi radian.

hence 180==pi rad
so 126 degree will correspond to 126pi/180 radian

so your angle is essentially (63/90)pi radian or 0.7pi

so the euler expression is merely 5e^(0.7pi)

3. Ah ok what about secend question ?

4. The definition of a logarithm of a complex number is given by

ln(z1)= ln(r)+ i(theta)

so ln(z1) is simply ln(5)+i(0.7pi)

The result is very simple to obtain

as z1=re^iθ
Taking logarithms to the base "e"(i.e. "ln") on both sides,
ln(z1)=ln{5(e^iθ )}
Using the property of logarithms,

ln(z1)= ln(5) + ln(e^iθ)
=ln(5) + iθln(e)
but ln(e)=1
so ln(z1)=ln5 + iθ
just plug in theta.

5. WOW you are a mathematical genius thanks for help