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Question: If \( \left| z \right| = 1, \) prove that \( \dfrac{{z - 1}}{{z + 1}}(z \ne - 1), \) is a pure imagi...

If ∣z∣=1,\left| z \right| = 1, prove that z−1z+1(z≠−1),\dfrac{{z - 1}}{{z + 1}}(z \ne - 1), is a pure imaginary number. What will you conclude if z=1z = 1 ?

Explanation

Solution

Hint : A number that is expressed in terms of the square root of negative number. An imaginary number is a complex number that can be return as a real number multiplied by the imaginary unit which is defined by its property l2=−1{l^2} = - 1 the square of an imaginary number b1{b_1} is −b2- {b^2} , for example, 5i5i is an imaginary number and its square is −25- 25

Complete step-by-step answer :
A part of a conditional statement after then, for example, the conclusion of “if a line is horizontal then the line has slope 00 is the line that has slope 00 ”
Let, z=x+iyz = x + iy then ∣z∣2=x2+y2{\left| z \right|^2} = {x^2} + {y^2} therefore, the condition ∣z∣=1\left| z \right| = 1 is equivalent to,
We know that ∣z∣=1\left| z \right| = {1_{}}
⇒x2+y2=1\Rightarrow {x^2} + {y^2} = 1
Now, according to the equation
z−1z+1=x+iy−1x+iy+1\dfrac{{z - 1}}{{z + 1}} = \dfrac{{x + iy - 1}}{{x + iy + 1}}
Separating both side
=(x−1+iy)(x+1−iy)(x+1+iy)(x+1−iy) =(x2+y2−1)+2iy(x+1)2+y2  = \dfrac{{\left( {x - 1 + iy} \right)\left( {x + 1 - iy} \right)}}{{\left( {x + 1 + iy} \right)\left( {x + 1 - iy} \right)}} \\\ = \dfrac{{({x^2} + {y^2} - 1) + 2iy}}{{{{(x + 1)}^2} + {y^2}}} \\\
=2iy(x+1)2+y2= \dfrac{{2iy}}{{{{\left( {x + 1} \right)}^2} + {y^2}}}
Hence, z−1z+1\dfrac{{z - 1}}{{z + 1}} is purely imaginary
When ∣z∣=1\left| z \right| = 1
Provided z≠−1z \ne - 1
when z=1z = 1 ,
we have z−1z+1=0\dfrac{{z - 1}}{{z + 1}} = 0
Now, recall that according to the definition that 0 is a pure imaginary number. Since, the point that is imaginary number 0 which corresponds to z=0z = 0 lies on both the real and imaginary axis. So in this case also z−1z+1\dfrac{{z - 1}}{{z + 1}} is a pure imaginary number.
So, the correct answer is “z−1z+1\dfrac{{z - 1}}{{z + 1}} is a pure imaginary number”.

Note : An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2=−1{i^2} = - 1 . The square of the imaginary number of bibi is −b2- {b^2} . Imaginary numbers are also called complex numbers, and are used in real life applications, such as electricity, as well as quadratic equations.