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Chapter 2: Q24P (page 52)

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

Short Answer

Expert verified

The value for $\frac{z}{z}$ is given below.

$\frac{z}{z} = \frac{- 5 - 12 i}{13} \frac{z}{z} = \frac{x^{2} + 2 x y i - y^{2}}{x^{2} + y^{2}}$

Step by step solution

01

Given Information

The complex number is $\frac{z}{\overset{炉}{z}}$ .

02

Definition of the Complex number.

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

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

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

03

Find the value.

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

Find the values for $2 - 3 i$ .

$\frac{z}{\overset{炉}{z}} = \frac{2 - 3 i}{2 + 3 i} \frac{2 - 3 i}{2 + 3 i} 脳 \frac{2 - 3 i}{2 - 3 i} = \frac{(2 - 3 i)}{4 - {(3 i)}^{2}} = \frac{4 - 12 i - 9}{4 = 9} = \frac{- 5 - 12 i}{13}$

Find the values for $x + i y$ .

$\frac{z}{\overset{炉}{z}} = \frac{x + y i}{z = x - y i} \frac{x = y i}{x - y i} 脳 \frac{x + y i}{x + y i} = \frac{{(x + y i)}^{2}}{x^{2} - {(y i)}^{2}} = \frac{x^{2} + 2 x y i - y^{2}}{x^{2} + y^{2}}$

The value for $\frac{z}{\overset{炉}{z}}$ is given below:

$\frac{z}{\overset{炉}{z}} = \frac{- 5 - 12 i}{13} \frac{z}{\overset{炉}{z}} = \frac{x^{2} + 2 x y i - y^{2}}{x^{2} = y^{2}}$

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Most popular questions from this chapter

Solve for all possible values of the real numbers xand y in the following equations $\frac{x + i y}{x - i y} = - i$ .

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

$\frac{2 i - 1}{i - 2}$ .

Express the following complex numbers in the $x + i y$ form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others鈥攖ry it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

10.role="math" localid="1653071800850" $e^{i 蟺} + e^{- i 蟺}$

Show from the power series (8.1) that $e^{z_{1}} 路 e^{z_{2}} = e^{z_{1} + z_{2}}$ .

Solve for all possible values of the real numbers x and y in the following equations. $| 1 - (x + i y) | = x + i y$

See all solutions

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