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Asked Oct 15, 2019
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2
then z can
purely imaginary
15. Prove that if z2
only be purely real or
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2 then z can purely imaginary 15. Prove that if z2 only be purely real or

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Expert Answer

Step 1

Let us take z =...

Here, we need to prove that if z2 = 22 then z can only be purely real or purely
imaginary. So, z
= x + iy and z = x - iy.
_
So, we have
=>(x iy)
= x2 (iy)2 2x x iy x2 (-iy)2 2x x (-iy)
(Since (a b)2 = a2 + b2 + 2ab)
=> x2- y2 2ixy x2 - y2 - 2ixy (Since i2 =-1)
=> x2- y22ixy (x2 - y2 - 2ixy) 0
=>4ixy 0
=> 0 4ixy = 0 i x 0
(x- iy)2
help_outline

Image Transcriptionclose

Here, we need to prove that if z2 = 22 then z can only be purely real or purely imaginary. So, z = x + iy and z = x - iy. _ So, we have =>(x iy) = x2 (iy)2 2x x iy x2 (-iy)2 2x x (-iy) (Since (a b)2 = a2 + b2 + 2ab) => x2- y2 2ixy x2 - y2 - 2ixy (Since i2 =-1) => x2- y22ixy (x2 - y2 - 2ixy) 0 =>4ixy 0 => 0 4ixy = 0 i x 0 (x- iy)2

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