Is the complex number z=e^2e^(1+i pi) pure imaginary? Is it real pure?Write its imaginary part, its real part, its module and argument.

ankarskogC

ankarskogC

Answered question

2021-09-08

Is the complex number z=e2e1+iπ pure imaginary? Is it real pure?
Write its imaginary part, its real part, its module and argument. Write its complex conjugate.
Calculate and write the result in binomial form.

Answer & Explanation

toroztatG

toroztatG

Skilled2021-09-09Added 98 answers

Step 1
Given complex number,
z=e2e1+iπ
This can be written as
z=e2+1+iπ
=e3+iπ
Since
z=x+iy
where
x=real part
y=imaginary part
Step 2
Then,
x+iy=e3+iπ
x+iy=e3eiπ
x+iy=e3[cosπ+isinπ]
x+iy=e3[1+i×0]
x+iy=e3(1)
x+iy=e3
On comparing both sides
x=e3,y=0
Therefore,
real part=e3
imaginary part=0
Step 3
Now,
|z|=|e2e1+iπ|
=|e2||e1+iπ|
=e2|eeiπ|
=e3|1|
=e3
therefore , modulo = e3
Find argument.
Argument =tan1(yx)=tan1(0e3)=tan1(0)=π
Step 4
complex conjugate of z,
Since , find in above
z=e2e1+iπ
and
x+iy=e3
Then,
Conjugate is
xiy=e3
Step 5
given,
z23iy
Then,
x+iy23i(y)=xy+iy223i×2+3i2+3i
=(xy+iy2)(2+3i)4+9i2
=(xy+iy2)(2+3i)49

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