Chapter 2: Q23P (page 52)

Find each of the following in rectangular $a + z b$ form if $z = 2 - {}_{t}3$ ; if $z = x + {}_{z}y$ .
$\frac{1 + z}{1 - z}$ .

Short Answer

Expert verified

The value for $\frac{1 + z}{1 - z}$ is given below:

$\frac{1 + z}{1 - z} = \frac{- 6 - 3 i}{5} \frac{1 + z}{1 - z} = \frac{(1 - x^{2}) + 2 y i - y^{2}}{(1 - x^{2}) + y^{2}}$

Step by step solution

Given Information

The complex number is $\frac{1 + z}{1 - z}$ .

Definition of the Complex number

Complex number z can be written in the form of the real and imaginary numbers as:

$z = a + {}_{z}b$

Here z is the complex number, and a and b are real numbers.

Find the value.

$\frac{1 + z}{1 = z}$ The complex number is $\frac{1 + z}{1 - z}$ .

Find the values for 2-3i .

$\frac{1 + z}{1 - z} = \frac{3 - 3 i}{- 1 + 3 i} = \frac{3 - 3 i}{- 1 + 3 i} \times \frac{- 1 - 3 i}{- 1 - 3 i} = \frac{- 3 - 9 i + 3 i (3 i^{2})}{1 - {(3 i)}^{2}} = \frac{- 6 - 3 i}{5}$

Find the values for $x + i y$ .

$\frac{1 + z}{1 - z} = \frac{1 + x = y i}{1 - x - y i} \frac{1 + x = y i}{1 - x - y i} = \frac{(1 + x) + y i (1 + x) + y i}{(1 - x) - y i (1 - x) - y i} \times \frac{(1 - x) + y i}{(1 - x) + y i} = \frac{(1 - x^{2}) + y i + x y i = y i - x y i - y^{2}}{{(1 - x)}^{2} - {(i y)}^{2}} = \frac{(1 - x^{2}) + y i - y^{2}}{{(1 - x)}^{2} + y^{2}}$

The value for $\frac{1 + z}{1 - z}$ is given below:

$\frac{1 + z}{1 - z} = \frac{- 6 - 3 i}{5} \frac{1 + z}{1 - z} = \frac{(1 - x^{2}) + 2 y i - y^{2}}{{(1 - x)}^{2} + y^{2}}$

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Find each of the following in rectangular $a + z b$ form if $z = 2 - {}_{t}3$ ; if $z = x + {}_{z}y$ .
$\frac{1 + z}{1 - z}$ .

Short Answer

Step by step solution

Given Information

Definition of the Complex number

Find the value.

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Find each of the following in rectangular a+zb form if z=2-3t; if z=x+yz.1+z1-z.

Short Answer

Step by step solution

Given Information

Definition of the Complex number

Find the value.

One App. One Place for Learning.

Most popular questions from this chapter

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Find each of the following in rectangular $a + z b$ form if $z = 2 - {}_{t}3$ ; if $z = x + {}_{z}y$ .
$\frac{1 + z}{1 - z}$ .