MTH120 College Algebra Chapter 2.4

 2.4 Complex Numbers

When you have a square root of a negative number, you can express it as a multiple of the imaginary unit "i." The imaginary unit "i" is defined as $\sqrt{-1}$. Here's how to express square roots of negative numbers as multiples of "i":

1. Understand the Square Root of -1:

   • Recognize that $\sqrt{-1}$ is represented as "i."

2. Express the Square Root of a Negative Number:

   • Suppose you have $\sqrt{-�}$, where $�$ is a positive real number. You can express it as $\sqrt{�}\cdot \sqrt{-1}$.

3. Replace $\sqrt{-1}$ with "i":

   • Replace $\sqrt{-1}$ with "i" to get $\sqrt{�}\cdot �$.

4. Final Expression:

   • The square root of $-�$ is $\sqrt{�}�$.

Here are a couple of examples:

Example 1: Express $\sqrt{-4}$ as a multiple of "i."

1. Recognize that $\sqrt{-1}$ is "i."

2. Express $\sqrt{-4}$ as $\sqrt{4}\cdot \sqrt{-1}$.

3. Replace $\sqrt{-1}$ with "i" to get $2�$.

So, $\sqrt{-4}=2�$.

Example 2: Express $\sqrt{-9}$ as a multiple of "i."

1. Recognize that $\sqrt{-1}$ is "i."

2. Express $\sqrt{-9}$ as $\sqrt{9}\cdot \sqrt{-1}$.

3. Replace $\sqrt{-1}$ with "i" to get $3�$.

So, $\sqrt{-9}=3�$.

In both cases, you've expressed the square roots of negative numbers as multiples of "i." This is a common practice in complex number arithmetic and allows for the handling of numbers that involve imaginary units.

Imaginary and complex numbers are extensions of the real number system that allow us to work with numbers that cannot be represented by traditional real numbers. They are essential in various fields of mathematics, engineering, and science. Here's an overview of imaginary and complex numbers:

1. Imaginary Numbers (i):

• Imaginary numbers are a subset of complex numbers.
• They are represented by the symbol "i," where $�=\sqrt{-1}$.
• Imaginary numbers arise when you take the square root of a negative real number.
• For example, $\sqrt{-1}$ is an imaginary number and is denoted as "i."

2. Complex Numbers:

• Complex numbers are numbers that have both a real part and an imaginary part.
• They are represented in the form $�+��$, where $�$ is the real part, and $�$ is the imaginary part.
• $�$ and $�$ are real numbers.
• Complex numbers extend the real number system to include numbers that cannot be expressed as real numbers.
• Complex numbers are used to solve equations that have no real solutions, such as $\sqrt{-�}$.

3. Real and Imaginary Parts:

• In the complex number $�+��$:
  • $�$ is the real part, and it represents the real number component of the complex number.
  • $��$ is the imaginary part, and it represents the imaginary number component of the complex number.

4. Arithmetic with Complex Numbers:

• You can perform arithmetic operations (addition, subtraction, multiplication, division) with complex numbers.
• For example, to add two complex numbers $�+��$ and $�+��$, you add their real parts separately (i.e., $�+�$) and their imaginary parts separately (i.e., $�+�$) to get the sum $(�+�)+(�+�)�$.

5. Complex Plane:

• Complex numbers can be represented graphically on the complex plane.
• The horizontal axis represents the real part, and the vertical axis represents the imaginary part.
• The complex plane allows for geometric interpretations of complex numbers.

6. Conjugate of a Complex Number:

• The conjugate of a complex number $�+��$ is denoted as $\overline{�+��}$ or $�-��$.
• The conjugate of a complex number has the same real part but the opposite sign for the imaginary part.
• For example, the conjugate of $3+4�$ is $3-4�$.

7. Complex Conjugate in Division:

• Dividing a complex number by its conjugate results in a real number.
• For example, $\frac{�+��}{�-��}=\frac{(�+��)(�+��)}{(�-��)(�+��)}=\frac{{�}^2+{�}^2}{{�}^2+{�}^2}=1$, where $�$ and $�$ are real numbers.

Complex numbers are powerful tools in mathematics and engineering, particularly in fields like electrical engineering, signal processing, and quantum mechanics, where they are used to represent and manipulate quantities with both real and imaginary components.

To express an imaginary number in the standard form of a complex number, you'll typically have an imaginary part represented by "i" and a real part, which is often 0. In the standard form of a complex number $�+��$, the real part is $�$, and the imaginary part is $�$.

Here are the steps to express an imaginary number in standard complex form:

Step 1: Start with the imaginary number, which is in the form $��$, where $�$ is a real number.

Step 2: Identify the real part and the imaginary part. In the case of an imaginary number, the real part is 0, and the imaginary part is $�$.

Step 3: Write the complex number in standard form $�+��$, where $�$ is the real part (which is 0 in this case) and $�$ is the imaginary part.

For example, if you have the imaginary number $3�$, you can express it in standard complex form as:

$0+3�$

So, $3�$ in standard complex form is $0+3�$. The real part is 0, and the imaginary part is 3.

To express an imaginary number in standard form, you typically write it as a complex number in the form $�+��$, where $�$ is the real part and $�$ is the coefficient of the imaginary unit "i." Here's how to do it:

1. Start with the imaginary number, which is in the form $��$, where $�$ is a real number.

2. Identify the real part and the imaginary part. In the case of an imaginary number, the real part is 0, and the imaginary part is $�$.

3. Write the complex number in standard form $�+��$, where $�$ is the real part (which is 0 in this case) and $�$ is the coefficient of the imaginary unit "i."

For example, let's express the imaginary number $4�$ in standard form:

1. Start with $4�$.

2. The real part is 0, and the imaginary part is $4$.

3. Write it in standard form: $0+4�$.

So, $4�$ in standard form is $0+4�$. The real part is 0, and the imaginary part is 4.

Plotting a complex number on the complex plane involves using a Cartesian coordinate system where the horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part. Here are the steps to plot a complex number on the complex plane:

1. Understand the Complex Plane:

   • The complex plane is similar to the Cartesian plane used for real numbers, but it's used for complex numbers.
   • The horizontal axis is the real axis (often labeled "Re" or "x"), and the vertical axis is the imaginary axis (often labeled "Im" or "y").
   • The point where the axes intersect is the origin, which represents the complex number $0+0�$ (the real and imaginary parts are both zero).

2. Identify the Complex Number:

   • You'll have a complex number in the form $�+��$, where $�$ is the real part and $�$ is the imaginary part.

3. Plot the Complex Number:

   • Locate the point on the complex plane corresponding to the complex number $�+��$:
     • Move $�$ units to the right (positive direction) or left (negative direction) along the real axis.
     • Move $�$ units up (positive direction) or down (negative direction) along the imaginary axis.
   • The point where these movements intersect is the location of the complex number on the complex plane.

4. Label the Point:

   • You can label the point with the complex number, indicating both the real and imaginary parts.
   • For example, if you are plotting $3+4�$, you would move 3 units to the right along the real axis and 4 units up along the imaginary axis, and then label the point as $(3,4)$.

Let's work through an example:

Example: Plot the complex number $2-3�$ on the complex plane.

1. Understand the complex plane: It consists of a real axis and an imaginary axis, with the origin at (0, 0).

2. Identify the complex number: $�=2$ (real part) and $�=-3$ (imaginary part).

3. Plot the complex number:

   • Move 2 units to the right along the real axis.
   • Move 3 units down along the imaginary axis.

4. Label the point as $(2,-3)$.

So, the complex number $2-3�$ is plotted at the point (2, -3) on the complex plane.

To represent the components of a complex number on the complex plane, you'll need to identify both the real part and the imaginary part of the complex number and plot them as coordinates on the plane. Here are the steps:

1. Understand the Complex Plane:

   • The complex plane is a Cartesian coordinate system where the horizontal axis represents the real part (often labeled "Re" or "x") of the complex number, and the vertical axis represents the imaginary part (often labeled "Im" or "y").

2. Identify the Complex Number:

   • You'll have a complex number in the form $�+��$, where $�$ is the real part and $�$ is the imaginary part.

3. Plot the Real Part:

   • Move $�$ units to the right (if $�$ is positive) or to the left (if $�$ is negative) along the real axis.

4. Plot the Imaginary Part:

   • Move $�$ units up (if $�$ is positive) or down (if $�$ is negative) along the imaginary axis.

5. Label the Point:

   • You can label the point with the complex number, indicating both the real and imaginary parts.
   • For example, if you are representing the complex number $3+4�$, you would move 3 units to the right along the real axis and 4 units up along the imaginary axis, and then label the point as $(3,4)$.

Here's an example:

Example: Represent the complex number $1-2�$ on the complex plane.

1. Understand the complex plane: It consists of a real axis and an imaginary axis, with the origin at (0, 0).

2. Identify the complex number: $�=1$ (real part) and $�=-2$ (imaginary part).

3. Plot the real part:

   • Move 1 unit to the right along the real axis.

4. Plot the imaginary part:

   • Move 2 units down along the imaginary axis.

5. Label the point as $(1,-2)$.

So, the complex number $1-2�$ is represented as the point (1, -2) on the complex plane.

To plot a complex number on the complex plane, you'll need to identify its real and imaginary parts and then locate the point that corresponds to these components on the plane. Here are the steps:

1. Understand the Complex Plane:

   • The complex plane is a Cartesian coordinate system where the horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part. The origin is at (0, 0).

2. Identify the Complex Number:

   • You'll have a complex number in the form $�+��$, where $�$ is the real part and $�$ is the imaginary part.

3. Locate the Point:

   • Move $�$ units to the right (if $�$ is positive) or to the left (if $�$ is negative) along the real axis.
   • Move $�$ units up (if $�$ is positive) or down (if $�$ is negative) along the imaginary axis.

4. Label the Point:

   • You can label the point with the complex number, indicating both the real and imaginary parts.
   • For example, if you are plotting the complex number $3+4�$, you would move 3 units to the right along the real axis and 4 units up along the imaginary axis, and then label the point as $(3,4)$.

Let's work through an example:

Example: Plot the complex number $-2+3�$ on the complex plane.

1. Understand the complex plane: It consists of a real axis (horizontal) and an imaginary axis (vertical), with the origin at (0, 0).

2. Identify the complex number: $�=-2$ (real part) and $�=3$ (imaginary part).

3. Locate the point:

   • Move 2 units to the left along the real axis (since $�$ is negative).
   • Move 3 units up along the imaginary axis (since $�$ is positive).

4. Label the point as $(-2,3)$.

So, the complex number $-2+3�$ is represented by the point (-2, 3) on the complex plane.

Adding and subtracting complex numbers is similar to adding and subtracting real numbers, but you need to keep track of both the real and imaginary parts separately. To add or subtract two complex numbers $�+��$ and $�+��$, follow these steps:

For Addition: Add the real parts together and add the imaginary parts together:

$(�+��)+(�+��)=(�+�)+(�+�)�$

For Subtraction: Subtract the real parts and subtract the imaginary parts:

$(�+��)-(�+��)=(�-�)+(�-�)�$

Here are examples for both addition and subtraction:

Example 1 (Addition): Let's add $2+3�$ and $1-4�$:

$(2+3�)+(1-4�)=(2+1)+(3-4)�=3-�$

So, $2+3�$ added to $1-4�$ equals $3-�$.

Example 2 (Subtraction): Now, let's subtract $5+2�$ from $7-3�$:

$(7-3�)-(5+2�)=(7-5)+(-3-2)�=2-5�$

So, $7-3�$ minus $5+2�$ equals $2-5�$.

Just remember to keep the real and imaginary parts separate when performing addition and subtraction with complex numbers.

To find the sum or difference of two complex numbers, you'll add or subtract their real parts separately from their imaginary parts separately. Here are the steps for both addition and subtraction:

For Addition:

1. Add the real parts of the complex numbers.
2. Add the imaginary parts of the complex numbers.

For Subtraction:

1. Subtract the real parts of the complex numbers.
2. Subtract the imaginary parts of the complex numbers.

Let's work through examples for both addition and subtraction:

Example 1 (Addition): Find the sum of $2+3�$ and $1-4�$.

1. Add the real parts: $2+1=3$.
2. Add the imaginary parts: $3�-4�=-�$.

So, the sum of $2+3�$ and $1-4�$ is $3-�$.

Example 2 (Subtraction): Find the difference between $7-3�$ and $5+2�$.

1. Subtract the real parts: $7-5=2$.
2. Subtract the imaginary parts: $-3�-2�=-5�$.

So, the difference between $7-3�$ and $5+2�$ is $2-5�$.

In both cases, you keep the real and imaginary parts separate when performing addition and subtraction with complex numbers.

Adding and subtracting complex numbers involves working with both their real and imaginary parts separately. Here are the steps for both addition and subtraction:

For Addition:

1. Add the real parts of the complex numbers.
2. Add the imaginary parts of the complex numbers.

For Subtraction:

1. Subtract the real parts of the complex numbers.
2. Subtract the imaginary parts of the complex numbers.

Let's work through examples for both addition and subtraction:

Example 1 (Addition): Add $2+3�$ and $1-4�$.

1. Add the real parts: $2+1=3$.
2. Add the imaginary parts: $3�-4�=-�$.

So, $2+3�$ added to $1-4�$ equals $3-�$.

Example 2 (Subtraction): Subtract $5+2�$ from $7-3�$.

1. Subtract the real parts: $7-5=2$.
2. Subtract the imaginary parts: $-3�-2�=-5�$.

So, $7-3�$ minus $5+2�$ equals $2-5�$.

In both examples, you handle the real and imaginary parts separately when performing addition and subtraction with complex numbers.

To multiply complex numbers, you'll use the distributive property and apply the rules of multiplying real and imaginary numbers. Here are the steps:

Let's say you have two complex numbers: $�+��$ and $�+��$.

1. Apply the Distributive Property:

   • Multiply the first term of the first complex number by both terms of the second complex number.
   • Multiply the second term of the first complex number by both terms of the second complex number.

   This results in four products:

   $(�+��)(�+��)=��+���+���+��{�}^2$

2. Combine Like Terms:

   • Remember that ${�}^2=-1$, so $�{�}^2=-�$.
   • Combine the real parts ($��-��$) and the imaginary parts ($��+��$):

   $(�+��)(�+��)=(��-��)+(��+��)�$

This is the product of the two complex numbers $�+��$ and $�+��$.

Example: Let's multiply $2+3�$ and $1-4�$:

$(2+3�)(1-4�)$

Apply the distributive property:

$=2\cdot 1+2\cdot (-4�)+3�\cdot 1+3�\cdot (-4�)$

Simplify the products:

$=2-8�+3�-12{�}^2$

Since ${�}^2=-1$, replace ${�}^2$ with $-1$:

$=2-8�+3�+12$

Combine like terms:

$=14-5�$

So, $(2+3�)(1-4�)=14-5�$.

That's the product of the two complex numbers.

Multiplying a complex number by a real number is straightforward. To do this, you simply distribute the real number to both the real and imaginary parts of the complex number. Here are the steps:

Let $�$ be a real number, and let $�+��$ be a complex number.

To multiply the complex number $�+��$ by the real number $�$, you distribute $�$ to both the real and imaginary parts of the complex number:

$$\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">�</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">�</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}$$

So, the result is a complex number with the real part $�\cdot �$ and the imaginary part $�\cdot �$.

Here's an example:

Let's multiply the complex number $3+2�$ by the real number $4$:

$$\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}$$

So, $4\cdot (3+2�)=12+8�$.

That's the result of multiplying the complex number $3+2�$ by the real number $4$.

Here are a couple of examples of multiplying a complex number by a real number:

Example 1: Let's multiply the complex number $2-3�$ by the real number $-4$:

$$<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}$$

So, $-4\cdot (2-3�)=-8-12�$.

Example 2: Now, multiply the complex number $1+2�$ by the real number $3$:

$$\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}$$

So, $3\cdot (1+2�)=3+6�$.

In both examples, we distributed the real number to both the real and imaginary parts of the complex number to obtain the result.

To multiply two complex numbers together, you'll apply the distributive property and use the fact that ${�}^2=-1$. Here's how to do it:

Let's say you have two complex numbers: $�+��$ and $�+��$.

1. Apply the Distributive Property:

   • Multiply the first term of the first complex number by both terms of the second complex number.
   • Multiply the second term of the first complex number by both terms of the second complex number.

   This results in four products:

   $(�+��)(�+��)=��+���+���+��{�}^2$

2. Simplify the Products Involving $�$:

   • Remember that ${�}^2=-1$. Replace ${�}^2$ with $-1$ in the expression:

   $(�+��)(�+��)=��+���+���+��(-1)$

3. Combine Like Terms:

   • Combine the real parts ($��-��$) and the imaginary parts ($��+��$):

   $(�+��)(�+��)=(��-��)+(��+��)�$

This is the product of the two complex numbers $�+��$ and $�+��$.

Example: Let's multiply $2+3�$ and $1-4�$:

$(2+3�)(1-4�)$

Apply the distributive property:

$=2\cdot 1+2\cdot (-4�)+3�\cdot 1+3�\cdot (-4�)$

Simplify the products:

$=2-8�+3�-12{�}^2$

Since ${�}^2=-1$, replace ${�}^2$ with $-1$:

$=2-8�+3�+12$

Combine like terms:

$=14-5�$

So, $(2+3�)(1-4�)=14-5�$.

That's the product of the two complex numbers.

To divide one complex number by another, you'll need to use a process that rationalizes the denominator. This is similar to dividing two fractions. Here are the steps:

Let's say you want to divide the complex number $�+��$ by $�+��$.

1. Rationalize the Denominator:

   • Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $�+��$ is $�-��$.

   $\frac{�+��}{�+��}\cdot \frac{�-��}{�-��}$

2. Apply the Distributive Property:

   • Multiply the numerators and the denominators.

   $\frac{(�+��)(�-��)}{(�+��)(�-��)}$

3. Simplify the Products:

   • Use the distributive property to multiply the numerators and denominators.

   $\frac{(��+���-��{�}^2-���)}{({�}^2+{�}^2{�}^2)}$

4. Use the Fact ${�}^2=-1$:

   • Remember that ${�}^2=-1$. Replace ${�}^2$ with $-1$ in the expression.

   $\frac{(��+���+��-��)}{({�}^2-{�}^2)}$

5. Combine Like Terms:

   • Combine the real and imaginary parts in both the numerator and the denominator.

   $\frac{(��+��)+(��+��)�}{({�}^2-{�}^2)}$

This is the result of dividing the complex number $�+��$ by $�+��$.

Example: Let's divide $(2+3�)$ by $(1-4�)$:

$\frac{2+3�}{1-4�}$

Rationalize the denominator:

$\frac{(2+3�)(1+4�)}{(1-4�)(1+4�)}$

Apply the distributive property:

$\frac{2+8�+3�+12{�}^2}{1^2-(4�{)}^2}$

Use ${�}^2=-1$:

$\frac{2+8�+3�-12}{1-16{�}^2}$

Combine like terms:

$\frac{-10+11�}{1+16}$

Simplify further:

$\frac{-10+11�}{17}$

So, $\frac{2+3�}{1-4�}=\frac{-10+11�}{17}$.

That's the result of dividing the two complex numbers.

The complex conjugate of a complex number is formed by changing the sign of its imaginary part while keeping the real part unchanged. In other words, if you have a complex number $�+��$, its complex conjugate is $�-��$, where $�$ and $�$ are real numbers.

Here are the key steps to find the complex conjugate:

1. Start with a complex number in the form $�+��$, where $�$ and $�$ are real numbers.
2. Change the sign of the imaginary part. If it's positive, make it negative, and if it's negative, make it positive.
3. Keep the real part unchanged.

For example, let's find the complex conjugate of $3+2�$:

1. Start with $3+2�$.
2. Change the sign of the imaginary part: $3-2�$.

So, the complex conjugate of $3+2�$ is $3-2�$.

The complex conjugate is often denoted by adding a bar or star over the variable, like $\overline{�}$ or ${�}^{\ast }$, where $�$ represents the original complex number. It is a useful concept in various areas of mathematics, including complex number arithmetic and the analysis of complex functions.

Here are a couple of examples of finding the complex conjugate of a complex number:

Example 1: Find the complex conjugate of $4+7�$.

Start with $4+7�$, and change the sign of the imaginary part:

Complex conjugate: $4-7�$

So, the complex conjugate of $4+7�$ is $4-7�$.

Example 2: Find the complex conjugate of $-2�$.

Start with $-2�$, and change the sign of the imaginary part:

Complex conjugate: $2�$

So, the complex conjugate of $-2�$ is $2�$.

Example 3: Find the complex conjugate of $-3$.

In this case, there is no imaginary part (it's just a real number), so the complex conjugate remains the same:

Complex conjugate: $-3$

So, the complex conjugate of $-3$ is $-3$.

In each example, we changed the sign of the imaginary part (if it existed) while keeping the real part unchanged to find the complex conjugate.

To divide one complex number by another, you'll use the process of rationalizing the denominator, similar to dividing fractions. Here are the steps to divide a complex number $�+��$ by another complex number $�+��$:

1. Rationalize the Denominator:

   • Multiply both the numerator and denominator by the complex conjugate of the denominator, which is $�-��$.

   $\frac{�+��}{�+��}\cdot \frac{�-��}{�-��}$

2. Apply the Distributive Property:

   • Multiply the numerators and denominators.

3. Simplify the Products:

   • Use the fact that ${�}^2=-1$ to simplify $��\cdot (-��)$ to $-{�}^2{�}^2$.

4. Combine Like Terms:

   • Combine the real parts in the numerator and denominator separately from the imaginary parts.

5. Express the Result:

   • The result will be a complex number in the form $�+��$, where $�$ is the real part, and $�$ is the imaginary part.

Here's an example:

Example: Divide $(2+3�)$ by $(1-4�)$.

1. Rationalize the denominator:

$\frac{(2+3�)(1-4�)}{(1-4�)(1+4�)}$

2. Apply the distributive property:

$\frac{2-8�+3�-12{�}^2}{1-16{�}^2}$

3. Use ${�}^2=-1$ to simplify:

$\frac{2-8�+3�+12}{1-16(-1)}$

4. Combine like terms:

$\frac{14-5�}{1+16}$

5. Simplify further:

$\frac{14-5�}{17}$

So, $(2+3�)$ divided by $(1-4�)$ is $\frac{14-5�}{17}$.

That's the result of dividing the two complex numbers.

When simplifying powers of $�$, you can use the fact that ${�}^2=-1$ and extend it for higher powers of $�$. Here's how you simplify various powers of $�$:

1. ${�}^0=1$

   • Any non-zero number raised to the power of 0 is equal to 1.

2. ${�}^1=�$

   • This is just $�$ itself.

3. ${�}^2=-1$

   • This is a fundamental property of $�$.

4. ${�}^3=-�$

   • To find ${�}^3$, you can multiply ${�}^2$ by $�$: ${�}^3=({�}^2)\cdot �=(-1)\cdot �=-�$.

5. ${�}^4=1$

   • ${�}^4$ is ${�}^2$ raised to the power of 2, so it's $(-1{)}^2=1$.

6. ${�}^5=�$

   • You can use the cyclical pattern: ${�}^5={�}^{4+1}={�}^4\cdot {�}^1=1\cdot �=�$.

7. ${�}^6=-1$

   • ${�}^6$ is ${�}^2$ raised to the power of 3, so it's $(-1{)}^3=-1$.

The powers of $�$ repeat in a cyclical pattern every four powers, which is why ${�}^4$ equals 1, ${�}^8$ equals 1, and so on. You can simplify any power of $�$ by reducing it to one of the values above.

For example, if you need to simplify ${�}^{10}$, you can notice that ${�}^{10}={�}^{8+2}={�}^8\cdot {�}^2$. Since ${�}^8=1$ and ${�}^2=-1$, you get ${�}^{10}=1\cdot (-1)=-1$.

You can express ${�}^{35}$ in a more helpful way by using the cyclical pattern of powers of $�$. As mentioned earlier, the powers of $�$ repeat every four powers: ${�}^1=�$, ${�}^2=-1$, ${�}^3=-�$, and ${�}^4=1$. After that, the pattern repeats.

So, to express ${�}^{35}$ more conveniently, you can divide 35 by 4 to find the remainder:

${�}^{35}={�}^{4\cdot 8+3}$

Here, we divided 35 by 4, which gives a quotient of 8 and a remainder of 3. Now, use the pattern to simplify:

${�}^{35}={�}^{4\cdot 8+3}=({�}^4{)}^8\cdot {�}^3$

Since ${�}^4=1$ and ${�}^3=-�$, you get:

${�}^{35}=(1{)}^8\cdot (-�)=-�$

So, ${�}^{35}$ can be expressed as $-�$.

To evaluate $�$ when $�=2+�$ for the given quadratic equation $�={�}^2+3�+5$, you substitute the value of $�$ into the equation and perform the calculations:

$�=(2+�{)}^2+3(2+�)+5$

Now, let's simplify this step by step:

1. Square $2+�$: $(2+�{)}^2=4+4�+{�}^2$

2. Apply ${�}^2=-1$: $(2+�{)}^2=4+4�-1$

3. Simplify: $(2+�{)}^2=3+4�$

Now, substitute this result back into the original equation:

$�=(3+4�)+3(2+�)+5$

4. Distribute 3 in the second term: $�=3+4�+6+3�+5$

5. Combine like terms: $�=(3+6+5)+(4�+3�)$ $�=14+7�$

So, when $�=2+�$, the value of $�$ is $14+7�$.