Question

if z=(1-i) the z-1 is given by​

Answer 1

Answer:

Z-1 = -i

Step-by-step explanation:

Given : Z= 1 - i

Therefore, Z-1 = -i

Hope you understood

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Answer 2

Answer:

Let z=x+iy. Then ∣z∣

2

=x

2

+y

2

.

Therefore the condition ∣z∣=1 is equivalent to

x

2

+y

2

=1.

Now

z+1

z−1

=

x+iy+1

x+iy−1

=

(x+1+iy)(x+1−iy)

(x−1+iy)(x+1−iy)

=

(x+1)

2

+y

2

(x

2

+y

2

−1)+2iy

=

(x+1)

2

+y

2

2iy

by (1)

Hence

z+1

z−1

is purely imaginary when ∣z∣=1

provided z



=−1.

When z=1, we have

z+1

z−1

=0.

Now recall that according to the definition 2 given in $$\S2$$, 0 is a pure imaginary number, since the point 0 which corresponds to z=0 lies on both real and imaginary axes.

So in this case also,

z+1

z−1

is a pure imaginary number.