Find the principal argument and exponential form of z=i/(1+i) If z=x+iy is a complex number

Marvin Mccormick 2020-11-01 Answered
Find the principal argument and exponential form of z=i1+i If z=x+iy is a complex number
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Expert Answer

SkladanH
Answered 2020-11-02 Author has 80 answers

Rewrite the above complex number in the usual form z=x+iy as follows.
z=i1+i=i1+i1i1i [multiply and divide by the conjugate 1−i of 1+i]
=i(1i)(1+i)(1i)=ii212i2=(i+1)(1+1)=i+12=12+12i
Compare the complex number z=12+12i with z=x+iy and obtain x=12 and y=12.
The argument of a complex number z=x+iy is given by θ=tan1(yx)

Here, for z=12+12i

θtan1(1212)=tan1(1)=π4

Since both x=12 and y=12 are positive, the complex number z=12+12i lie in the 1st quadrant.

Hence, its principal argument is the same as θ=π4

The exponential form of a complex number z=x+iy with principal argument theta is given by z=reiθ where r=x2+y2

For z=12+12i the value r=x2+y2=(12)2+(12)2=12=12

Hence, the exponential form of z=12+12i is z=12+12i is z=12ei(π4)

Thus, the principal argument of z=i1+i is 0=π4 and the exponential form is z12ei(π4)

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