Complex Analysis

Objectives

Students will:

(1.) Define real numbers.

(2.) Define complex numbers.

(3.) Express irrational radicals in terms of the imaginary unit.

(4.) Write the various forms of complex numbers.

(5.) Simplify powers of the imaginary unit.

(6.) Add complex numbers.

(7.) Subtract complex numbers.

(8.) Multiply complex numbers.

(9.) Divide complex numbers.

(10.) Solve calculations involving complex numbers using the TI-84 / TI-84 Plus.

Skills

(1.) Prior knowledge.

(2.) Mathematical Reasoning.

(3.) Use of Technology.

Relate with English: number, real, complex, power, radical, standard, form, add, subtract, multiply, divide
Relate with Mathematics: number, real number, complex number, complex analysis, real analysis, complex variables, numbers, complex numbers, real part, imaginary part, pure imaginary number, imaginary unit, standard form, polar form, power, exponent, exponential form, sum, difference, product, quotient, add, subtract, multiply, divide, rationalize, radical, radicand

A real number is any rational or irrational number.

It includes all numbers that can be found on the real number line.

Some students may ask you the definitions of rational numbers and irrational numbers.
Refer them to this website: Numbers and Notations

A complex number is a number that can be expressed in the form of $a + bi$ where $a \:and\: b$ are real numbers, and $i$ is an imaginary number equal to the square root of $-1$.

All real numbers can be written as complex numbers. However, not all complex numbers can be written as real numbers. Why?

Say we have:
$ x^2 = 16 \\[3ex] x = \sqrt{16} \\[3ex] x = \pm 4 $ The principal or positive square root = $4$
The negative square root = $-4$

What about:
$ x^2 = -16 \\[3ex] x = \sqrt{-16} $
Ask students to give you the answer. They can put it in their calculators. Note their responses. Were there any values in their answers?
Teach them how to change to the complex number mode. Ask them to do it again. Do they see any values in their answers?

$ x = \sqrt{-16} \\[3ex] x = \pm 4i $
For "just" the square root of real numbers, we shall deal only with the principal square root. $ \sqrt{-16} \\[3ex] = \sqrt{-1} * \sqrt{16} ...\:Product \:Rule \:of\: Radicals \\[3ex] \sqrt{cd} = \sqrt{c} * \sqrt{d} ...\:Product \:Rule \:of\: Radicals \\[3ex] = i * 4 \\[3ex] = 4i \\[3ex] where\: i = \sqrt{-1} $
Hmmmm....some students may ask why you should use the Product Rule of Radicals when one of the radicands is a negative number.
Some learned that the Product Rule of Radicals applies only if the radicands are positive real numbers.

Here is the thing:
Recall the Product Rule of Radicals:
$\sqrt[n]{cd} = \sqrt[n]{c} * \sqrt[n]{d}$
where:
$n$ is a positive integer
$c$ and $d$ are real numbers.

What about: $\sqrt[3]{-64}$?
It could be written as:
$ \sqrt[3]{-64} \\[3ex] = \sqrt[3]{-8 * 8} \\[3ex] = \sqrt[3]{-8} * \sqrt[3]{8} \\[3ex] = -2 * 2 \\[3ex] = -4 $
Yes, $\sqrt[3]{-64}$ = -4
So, $c$ and $d$ must not be positive real numbers.

We can also define complex numbers as the "even-th" roots (square root, fourth root, sixth root, eighth root, tenth root,...) of negative real numbers.
Ask students why it has to be "even-th" roots. Why not "odd-th" roots?
Hint: What is the cube root of $-8$?
Hint: What is the fourth root of $-16$?
Hint: What is the fifth root of $-625$?

Complex Numbers can be represented in three forms:
(1.) Standard form
(2.) Polar form
(3.) Exponential form

Standard Form of a Complex Number
The standard form of a complex number is written as:
$a + bi$
where:
$a$ and $b$ are real numbers
$a$ = real part of the complex number
$b$ = imaginary part of the complex number
$i$ = imaginary unit
$bi$ = pure imaginary number

Examples
(1.) $3 + 7i$
$a$ = real part = $3$
$b$ = imaginary part = $7$

(2.) $-3 + 10i$
$a$ = real part = $-3$
$b$ = imaginary part = $10$

(3.) $-3 - 12i$
$a$ = real part = $-3$
$b$ = imaginary part = $-12$

(4.) $3$
This means: $3 + 0i$
$a$ = real part = $3$
$b$ = imaginary part = $0$

Ask students if they noticed that they can write any real number as a complex number.

(5.) $-3$
This means: $-3 + 0i$
$a$ = real part = $-3$
$b$ = imaginary part = $0$

Ask students if they noticed they can write any real number as a complex number.

(6.) $7i$
This is a pure imaginary number. It is a complex number.
It can be written as: $0 + 7i$
$a$ = real part = $0$
$b$ = imaginary part = $7$

Ask students if they noticed they cannot write any complex number as a real number.

(7.) $-7i$
This is a pure imaginary number. It is a complex number.
It can be written as: $0 - 7i$
$a$ = real part = $0$
$b$ = imaginary part = $-7$

Ask students if they noticed they cannot write any complex number as a real number.
All real numbers can be written as complex numbers. However, not all complex numbers can be written as real numbers. You see the reasons?

$i = \sqrt{-1}$
$ i^2 = (\sqrt{-1})^2 \\[3ex] = -1 \\[3ex] $ $i^2 = -1$

This is what we did.

$ (\sqrt{-1})^2 \\[3ex] = (-1^\dfrac{1}{2})^2 \\[3ex] x^\dfrac{1}{2} = \sqrt{x} ...\: Fractional\: Exponents\: gives\: radicals \\[7ex] = 3^{\dfrac{1}{2} * 2} ...\: Law\:7...Exp \\[5ex] = 3^1 \\[3ex] = 3 \\[3ex] $ As you can see:
The square of the square root of a number is the number.
$(\sqrt{3})^2 = 3$
$ (\sqrt{3})^2 \\[3ex] = (3^\dfrac{1}{2})^2 \\[5ex] x^\frac{1}{2} = \sqrt{x} ...\: Fractional\: Exponents\: gives\: radicals \\[5ex] = 3^{\frac{1}{2} * 2} ...\: Law\:7...Exp \\[5ex] = 3^1 \\[3ex] = 3 \\[3ex] $
Similarly, the square root of the square of a number is the number.
$\sqrt{3^2} = 3$
$ \sqrt{3^2} \\[3ex] = (3^2)^\dfrac{1}{2} \\[5ex] = 3^{2 * \dfrac{1}{2}} ...\: Law\:5...Exp \\[5ex] = 3^1 \\[3ex] = 3 \\[3ex] $ Ask students if they agree with the postulate? Lol... bring it to Geometry ☺
If they agree, they should give reasons using the Law of Exponents...for Fractional Exponents.
If they disagree, they should give reasons.

Explain to students this common misconception
** Common Misconception **

$ (\sqrt{-1})^2 \\[3ex] = \sqrt{-1} * \sqrt{-1} \\[3ex] = \sqrt{-1 * -1} \\[3ex] = \sqrt{1} \\[3ex] = 1 \\[3ex] $ Notice that those who did it that way probably used the Product Rule of Radicals: $\sqrt{ab} = \sqrt{a} * \sqrt{b}$

Here is the thing again: ☺
Recall the Product Rule of Radicals:
$\sqrt[n]{cd} = \sqrt[n]{c} * \sqrt[n]{d}$
where:
$n$ is a positive integer
$c$ and $d$ are real numbers.

Since we have already established that $\sqrt{-1} = i$
Why do we need to use the Product Rule to find $i^2$?
Law of Exponents takes precedence before Product Rule. Remember PEMDAS
Ask students the meaning of PEMDAS.
PEMDAS is an acronym that stands for: Parenthesis; Exponents; Multiplication and Division (whichever comes first); Addition and Subtraction (whichever comes first).
In all our calculations, we have to express the square root of any negative number in terms of $i$ first.

$ i^3 = i^2 * i ...\: Law\;1...Exp \\[3ex] = -1 * i \\[3ex] = -i \\[3ex] $ $i^3 = -i$

$ i^4 = i^2 * i^2 ...\: Law\;1...Exp \\[3ex] = - 1 * -1 \\[3ex] = 1 \\[3ex] $ $i^4 = 1$

All other powers of $i$ can be evaluated using any of $i^2$, $i^3$, or $i^4$
But, what is the easiest one to use?
Ask students to say the easiest one to use. Why?

It's $i^4$ of course!
$i^4$ = 1
$1$ raised to any exponent gives $1$
So, it is much better to evaluate all other powers of $i$ based on $i^4$.

Some students may ask if they could use $i^2$
Of course, they can.
However, please note that: $i^2 = -1$
A negative base raised to an even exponent will give a positive result.
But, a negative base raised to an odd exponent will give a negative result.
You may say it in "reverse" and ask students to correct you if applicable.
In other words, say something wrong, so students can correct it.

Notable Notes for the Powers of $i$

(1.) A number is divisible by $4$ if the last two digits are divisible by $4$

(2.) A number is divisible by $2$ if it is an even number. This includes $0$

(3.) A positive base raised to an even exponent gives a positive result.

(4.) A positive base raised to an odd exponent gives a positive result.

(5.) A negative base raised to an even exponent gives a positive result.

(6.) A negative base raised to an odd exponent gives a negative result.

(7.) $(-1)$ raised to an even exponent gives $1$.

(8.) $(-1)$ raised to an odd exponent gives $-1$.

(9.) $(1)$ raised to an any exponent(even, odd, fractional, radical) gives $1$.

(10.) Any even number which is not divisible by $4$, when divided by $2$; gives an odd number.

This calculator will:
(1.) Calculate the $powers\: of\: i$
(2.) Add, subtract, multiply, and divide complex numbers.
(3.) Return the real part.
(4.) Return the imaginary part.
(5.) Return the answer in standard form.

• Using the TI-84 / 84 Plus Calculator for Complex Numbers Calculations
• All outputs/answers are in the standard form of complex numbers.
• Visit the link: TI-84 Calculator
• Click the "Evaluating Complex Arithmetic" folder
• Click the "Evaluating complex arithmetic" link
• Follow the directions as illustrated by the "blinking light"

This calculator will:
(1.) Simplify radicals with negative perfect square radicands.
(2.) Add, subtract, multiply, and divide radicals with negative perfect square radicands.
(3.) Calculate the $powers\: of\: i$
(4.) Add, subtract, multiply, and divide simple complex numbers.
(5.) Return the real part.
(6.) Return the imaginary part.
(7.) Return the answer in standard form.

• Using the Complex Numbers Calculator - First
• All outputs/answers are in the standard form of complex numbers.
• All outputs/answers are written as both integers and/or decimals; and integers and/or fractions.
• You can perform arithmetic operations on many radicals and/or complex numbers at a time.
• Radicals: Type $\sqrt{-16}$ as sqrt(-16)
• Radicals: Type $-\sqrt{-16}$ as -sqrt(-16) OR -1 * sqrt(-16)
• Add Radicals: Type $(7 + \sqrt{-25}) + (3 + \sqrt{-4})$ as (7 + sqrt(-25)) + (3 + sqrt(-4))
• Subtract Radicals: Type $(7 + \sqrt{-25}) - (3 + \sqrt{-4})$ as (7 + sqrt(-25)) - (3 + sqrt(-4))
• Multiply Radicals: Type $(16 + \sqrt{-25})(4 + \sqrt{-100})$ as (16 + sqrt(-25)) * (4 + sqrt(-100))
• Divide Radicals: Type $(16 + \sqrt{-25}) \div (2 + \sqrt{-4})$ as (16 + sqrt(-25)) / (2 + sqrt(-4))
• Powers of $i$: Type $(7i)^3$ as (7i)^3
• Powers of $i$: Type $(-7i)^3$ as (-7i)^3
• Powers of $i$: Type $i^{14379765324}$ as i^1379765324
• Powers of $i$: Type $i^{-14379765330}$ as i^-1379765330
• Add Complex Numbers: Type $(7 + 5i) + (3 + 2i)$ as (7 + 5i) + (3 + 2i)
• Subtract Complex Numbers: Type $(-7 + 5i) - (3 - 2i)$ as (-7 + 5i) - (3 - 2i)
• Multiply Complex Numbers: Type $(-16 + 5i)(4 - 10i)$ as (-16 + 5i) * (4 - 10i)
• Multiply Complex Numbers: Type $(-16 + 5i)^2$ as (-16 + 5i)^2
• Divide Complex Numbers: Type $(16 + 5i) \div (2 + 4i)$ as (16 + 5i) / (2 + 4i)
• Add, Subtract, Multiply, and Divide Radicals and Complex Numbers
• Put the parenthesis appropriately
• When there are several arithmetic operators, the calculators does the calculations according to the order of operations
• You may type integers, fractions, and/or decimals
• Type $\dfrac{-7 - 3i}{12i - 5} - \dfrac{-3i - 7}{5i + 12}$ as ((-7 - 3i) / (12i - 5)) - ((-3i - 7) / (5i + 12))
• Type $\left(\dfrac{5}{2} + \sqrt{-\dfrac{4}{25}}\right) - \left(-\dfrac{4i}{5}\right) * \left(\dfrac{3i}{10} - \dfrac{2}{5}\right)^2$ as ((5/2) + sqrt(-4/25)) - (-4i/5) * ((3i/10) - (2/5))^2

This calculator will:
(1.) Simplify radicals with negative radicands.
(2.) Add, subtract, multiply, and divide radicals with negative radicands.
(3.) Calculate the $powers\: of\: i$
(4.) Add, subtract, multiply, and divide complex numbers.
(5.) Return the answer in standard form.

To use the calculator, please:
(1.) Type your expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" expression in the textbox of the calculator.
(4.) Copy and paste the expression you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct expression you typed.
(7.) Review the answers. "At least" one of the answers is the one you probably need.

• Using the Complex Numbers Calculator - Second
• All outputs/answers are in the standard form of complex numbers.
• All outputs/answers are written as both integers and/or decimals; and integers and/or fractions.
• You can perform arithmetic operations on many radicals and/or complex numbers at a time.
• Radicals: Type $\sqrt{-7}$ as sqrt(-7)
• Radicals: Type $-\sqrt{-72}$ as -sqrt(-72) OR -1 * sqrt(-72)
• Add Radicals: Type $(7 + \sqrt{-25}) + (3 + \sqrt{-4})$ as (7 + sqrt(-25)) + (3 + sqrt(-4))
• Subtract Radicals: Type $(7 + \sqrt{-25}) - (3 + \sqrt{-4})$ as (7 + sqrt(-25)) - (3 + sqrt(-4))
• Multiply Radicals: Type $(16 + \sqrt{-25})(4 + \sqrt{-100})$ as (16 + sqrt(-25)) * (4 + sqrt(-100))
• Divide Radicals: Type $(16 + \sqrt{-25}) \div (2 + \sqrt{-4})$ as (16 + sqrt(-25)) / (2 + sqrt(-4))
• Powers of $i$: Type $(7i)^3$ as (7i)^3
• Powers of $i$: Type $(-7i)^3$ as (-7i)^3
• Powers of $i$: Type $i^{14379765324}$ as i^1379765324
• Powers of $i$: Type $i^{-14379765330}$ as i^-1379765330
• Add Complex Numbers: Type $(7 + 5i) + (3 + 2i)$ as (7 + 5i) + (3 + 2i)
• Subtract Complex Numbers: Type $(-7 + 5i) - (3 - 2i)$ as (-7 + 5i) - (3 - 2i)
• Multiply Complex Numbers: Type $(-16 + 5i)(4 - 10i)$ as (-16 + 5i) * (4 - 10i)
• Multiply Complex Numbers: Type $(-16 + 5i)^2$ as (-16 + 5i)^2
• Divide Complex Numbers: Type $(16 + 5i) \div (2 + 4i)$ as (16 + 5i) / (2 + 4i)
• Divide Complex Numbers: Type $(7 + \sqrt{3}i) \div (10 - 21i)$ as (7 + i * sqrt(3)) / (10 - 2i)
• Add, Subtract, Multiply, and Divide Radicals and Complex Numbers
• Put the parenthesis appropriately
• When there are several arithmetic operators, the calculators does the calculations according to the order of operations
• You may type integers, fractions, and/or decimals
• Type $\dfrac{-7 - 3i}{12i - 5} - \dfrac{-3i - 7}{5i + 12}$ as ((-7 - 3i) / (12i - 5)) - ((-3i - 7) / (5i + 12))
• Type $\left(\dfrac{5}{2} + \sqrt{-\dfrac{4}{25}}\right) - \left(-\dfrac{4i}{5}\right) * \left(\dfrac{3i}{10} - \dfrac{2}{5}\right)^2$ as ((5/2) + sqrt(-4/25)) - (-4i/5) * ((3i/10) - (2/5))^2

Simplify