Think Like A Stoic: Notes

Lesson 1: How To Live Like A Stoic

Zeno's Shipwreck

More than 2,000 years ago, a Phoenician merchant named Zeno of Citium was on board a merchant ship sailing in the Aegean Sea near Athens. The ship encountered a powerful storm and sunk with all of its cargo and many men aboard. But Zeno survived, making it ashore and arriving at Athens.

Once he recovered from the shock of this experience, he went to a bookshop, where he listened to the Memorabilia being read aloud. This is a composition written by the Greek statesman and writer Xenophon about the famous Athenian philosopher Socrates.

Zeno was so fascinated that he resolved then and there to study philosophy. He turned to the bookseller and asked, “Where can I find a philosopher?” The bookseller looked out onto the street and said, “There goes one!” The philosopher in question was Crates of Thebes. And Zeno became his student.

After a few years of studying with a number of other teachers, Zeno felt ready to begin his own school of practical philosophy, which became known as Stoicism.

Virtue Ethics

While standard ethics is focused on answering questions about actions (e.g., "Is abortion right or wrong?" or "Is human cloning right or wrong?"), virtue ethics, by contrast, focuses on our character. It asks, "How can I become a better person?" (It focuses on the character traits and virtues that make a person good rather than evaluating isolated actions).

Aristotle practically invented the approach of virtue ethics as it is understood in the West today. He thought that a eudaimonic life—that is, a life worth living—is one in which we try to become the best humans we can be. But he also contended that such a life requires many things that are not under our control, such as wealth, education, health, and even beauty.

Epicurus, in contrast, thought that the most important things in life were the pursuit of simple pleasures and the avoidance of pain, both physical and emotional. So, the Epicurean life consists of spending a lot of time with friends, reading and debating, all while accompanied by food and wine[

Stoicism is another type of virtue ethics, alongside Aristotelianism and Epicureanism. Indeed, the Stoics were major rivals of the two, and of other schools throughout the Hellenistic period. This is an ancient philosophy born of the need of a Phoenician merchant who was trying to process losing all he had after barely surviving a shipwreck. A sound philosophy of life can change your outlook for the better, as it did for Zeno of Citium

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The Premise of Stoicism 

Stoic philosophy is founded on one crucial premise and relies on two fundamental pillars to provide practical guidance through one’s life.

The premise is to live according to nature.
The Stoics figured that if the problem we’re facing is how to live a life worth living, then we should take seriously what sort of animal human beings actually are (what is our nature). Two ideas, according to the Stoics, are so important that we might want to organize our entire existence around them:

We are capable of reason, and

We are inherently social animals.

A capacity for reason doesn’t mean that we are always reasonable, of course. And being social doesn’t mean we cannot live in isolation. Instead, it means that we thrive in a social group, pursuing projects that are made possible by the fact that we live in a society.

From these two observations, the Stoics concluded that a life well lived is one in which we deploy reason for the improvement of society. Living according to nature means using our brains, as imperfect as our brains are, to make life on this planet better for everyone, and therefore for ourselves.

The Four Cardinal Virtues

We can achieve living a life well lived by relying on the two pillars of Stoic philosophy: the four cardinal virtues and the dichotomy of control.

Improving society, according to the Stoics, isn’t something that can be done from the top down—by imposing some kind of utopia on people who might or might not like your view of how things should be run. Instead, the world changes one person at a time, from the bottom up. And the only person you can change is yourself.

This positive personal change comes about by constantly practicing four virtues, or habits. They are: 1) practical wisdom, 2) courage, 3) justice, and 4) temperance.

Practical wisdom tells us the difference between what we can and cannot change. 

Courage is not just physical but first and foremost moral—the courage to stand up and do the right thing. 

Justice tells us the right thing to do. 

Temperance is the notion that we should do things in the right measure, neither overreacting to situations nor failing to do enough to correct things

The Dichotomy of Control

(The dichotomy of control divides all things in life into two categories: Things within our control and Things outside our control.)

The second pillar of Stoic philosophy—the dichotomy of control—was famously summarized by the slave-turned-teacher Epictetus at the beginning of his manual for a good life, The Enchiridion. While we can influence our body, property, reputation, and office, ultimately, they’re not under our power. Even the healthiest body can be struck by disease or accident. Your property can be taken away from you for a number of reasons. And your reputation can be ruined through gossip and other people’s malicious intent.

What if your coworker is being harassed by the boss? Practical wisdom tells you that this is a situation where you can intervene. You might not be able to change your boss’s attitude, but you may be able to improve the culture at your workplace and comfort your coworker. Justice, then, lies in standing up to the boss. Courage is what gives you the strength to do it. And temperance keeps your response to the boss within reasonable and useful limits[search_files_v2:1].

In contrast, opinion, motivation, desire, aversion, and other things of our own doing might be influenced by others, but they ultimately are our own responsibility. Other people might try to get you to change your judgments and opinions or make you adopt a different set of values, but the buck, so to speak, stops with you[search_files_v2:1].

If you take the dichotomy of control to heart, you’ll change your entire outlook on life. You’ll no longer concern yourself with the outcomes of your decisions but instead with their soundness. The outcome is not up to you, but the decision to do certain things rather than others certainly is. The Roman writer Cicero explained the Stoic position by considering an archer who is trying to hit a target. The archer can decide how assiduously to practice, which arrows and bow to select, and how to care for them. They also control their focus right up the moment they let go of the arrow. But once the arrow leaves the bow, nothing at all is under the archer’s control. A sudden gust of wind might deflect the best shot, or the target—say, an enemy soldier—might suddenly move. Hitting the target is what you’re after, so it’s what you pursue. But success or failure does not, in and of itself, make you a good or bad archer. This means that you should not attach your self-worth to the outcome but only to the attempt. Then, you will achieve what the ancients called ataraxia—the kind of inner tranquility that results from knowing you’ve done everything that was in your power to do

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Role Ethics

Another way to understand and practice Stoic philosophy is through what’s known as role ethics. The idea is that we all play a variety of roles in our lives—father, son, mother, daughter, boss, employee, and so forth—and that a life worth living involves balancing those roles in the most harmonious way possible

According to Epictetus, there are fundamentally three kinds of roles:

• our basic role as human beings and members of the human cosmopolis,
• roles that are given to us by circumstances, such as being someone’s daughter or son, and
• roles that we choose for ourselves, such as by having children or being someone’s friend.

The Stoics think that our role as a member of the human cosmopolis is the most important one. Ultimately, we need to work in concert to make sure that we provide a better present for all and a better future for coming generations. Within that broad guideline, we then need to balance all of our specific roles the best way possible. That will often entail compromises. How much of a compromise should you make, say, when you contemplate the simultaneous demands of your professional and family lives? This ties back to the cardinal virtues

The ancient Stoics had a trick to improving our character— the use of role models. They came up with a gallery of real and fictional role models whose virtues they used to straighten our crooked character. 

One of the most famous of these was Cato the Younger, who gave his life to oppose what he perceived as the tyranny of the Roman dictator Julius Caesar. Another was the mythical Odysseus, who endured 10 years of hardship and turned down the offer of immortality—twice—in order to come back home to his wife and son. And, of course, there is Socrates, whose philosophy inspired the Stoics, and whose death at the hands of the state made him the first martyr to the cause of wisdom.

Lesson 2: Stoicism From Heraclitus To Thoreau

To help situate Stoicism among its rivals and influencers, this lesson takes a look at the major Hellenistic schools of thought. It then examines Stoicism’s philosophical roots in Socrates and Heraclitus and its evolution through time.

THE HELLENISTIC PERIOD

Alexander the Great ruthlessly built the largest empire the Western world had ever seen by the
time he was 32. Then came the catastrophe that abruptly quashed his dreams: He died after a night and day of boozy partying with one of his admirals—or perhaps he was poisoned.

Regardless, the world that Alexander constructed with so much effort crumbled overnight. With no obvious or legitimate heir to replace him, a 40-year war erupted, all while Rome was asserting itself as a new power in the Mediterranean world.

In September of the year 31 BCE, a decisive naval confrontation took place at Actium, on the western coast of present-day Greece. This was between the forces of Caesar Octavian on one side and Mark Antony and Cleopatra on the other. Octavian won, ending the Roman civil war that had persisted since the death of Julius Caesar some 13 years earlier. Under the name of Caesar Augustus, Octavian became the first Roman emperor.

The timespan bracketed by the death of Alexander and the battle of Actium is known as the Hellenistic period. It saw the sudden flourishing of a number of philosophies, including Epicureanism, Cyrenaism, Cynicism, Peripateticism, Skepticism, and Stoicism. It may be no coincidence that the people of this era were interested in practical philosophy, given that their world had been turned upside down by events they had no hope of influencing, let alone controlling.

HELLENISTIC SCHOOLS OF THOUGHT

Lesson 3: The Stoic Garden: Physics, Ethics, Logic

HESIOD: WORKS AND DAYS

1. Hymn to Zeus

2. The Two Strifes

3. Pandora and the Jar

-Pandora's Box: The Story of the First Woman Created by the Gods

4. The Ages of Man

5. Fable of the Hawk & Nightingale

6. On Justice & Good Conduct

7. Agrarian Calendar, Farming & Fishing

8. Traditional Customs

9. Auspicious Days of Month

Greek Mythology Sources

The Iliad by Homer - around 8th century B.C.E.

The Odyssey by Homer - around 8th century B.C.E. 

Greek playwrights

Aeschylus: Known for plays like the Oresteia trilogy and Prometheus Bound.
Sophocles: Famous for his tragedies about Oedipus and Antigone.
Euripides: Author of Medea, The Bacchae, and The Trojan Women, among others.

Theogony by Hesiod - around 700 B.C.E. 

Works and Days by Hesiod - around 700 B.C.E.

The Bibliotheca by Apollodorus - 1st or 2nd century  C.E.

Metamorphoses by Ovid - around 8th century C.E.

Greek Deities & Family Trees

It's a confusing mess. Here's some attempt to organize but some details may be wrong. 

Here's a very loose generational list:

First Era

Primordial Deities 

Chaos (The Void)

Gaia (The Earth)

Tartarus (The Abyss)

Nyx (The Night)

Erebus (Darkness)

Uranus (The Sky) - Son and husband of Gaia

Second Era

Titans 

Cronus

Rhea

Oceanus

Tethys

Hyperion

Theia

Coeus

Phoebe

Lapetus

Crius

Mnemosyne

Themis

Third Era

The Olympian Gods

The Olympian gods, in Greek mythology, were the major deities who resided on Mount Olympus. There were traditionally twelve Olympians, though the specific members could vary slightly, with Hestia or Dionysus sometimes taking the twelfth spot.

12 Olympian Gods

Zeus (Roman Jupiter) - King of the Gods - God of the Sky and Thunder - Lightning Bolt and Eagle.

Hera (Roman Juno) -

Poseidon (Neptune) - God of the Sea, Earthquakes and Horses - Trident.

Demeter (Roman Ceres) - Goddess of the Harvest - torch, chariot with wings

Apollo (Roman Apollo) -

Artemis (Roman Diana) -

Ares (Roman Mars) -

Athena (Roman Minerva) - 

Hephaestus (Roman Vulcan) -

Aphrodite (Roman Venus) -

Hermes (Roman Mercury) - 

The twelfth being either:

Hestia (Roman Vesta) - 

or

Dionysus (Roman Liber) - 

Olympian Era Gods (Don't Live On Mount Olympus)

Hades (Pluto) - God of the Underworld - Helm of Darkness (helmet of invisibility) and Cerberus (Three headed dog. 

Notes: 

Era vs Generation - Most people seem to use the word "generation" instead of "era" when referring to the successions of power. This is very confusing and took me a while to understand. For instance, Atlas is referred to as a titan but he is actually a cousin of Zeus. Same with Prometheus. Therefore, I used the term era instead of generation. 

Wikipedia: Family tree of the Greek gods

Wikipedia: Twelve Olympians

Amortization Schedules

When you borrow money, you have to repay the amount of money you borrowed plus the interest the loan accumulates over the term of the loan. You repay the loan by making regular payments over the term of the loan. Each payment does two things: it pays all of the interest due on the loan at the time the payment is made and the remainder of the payment goes to paying down the loan amount. This process of gradually repaying the loan with periodic payments over the term of the loan is called amortization.

Amortization is a process by which the principal of a loan is extinguished over the course of an agreed-upon time period through a series of regular payments that go toward both the accruing interest and principal reduction. (the process of spreading out a loan or asset cost into equal payments over time. Each payment typically includes both interest and principal.) Two components make up the agreed-upon time component. 

1. Amortization Term. The amortization term is the length of time for which the interest rate and payment agreement between the borrower and the lender will remain unchanged. Thus, if the agreement is for monthly payments at a 5% fixed rate over five years, it is binding for the entire five years. Or if the agreement is for quarterly payments at a variable rate of prime plus 2% for three years, then interest is calculated on this basis throughout the three years.
2. Amortization Period. The amortization period is the length of time it will take for the principal to be reduced to zero. For example, if you agree to pay back your car loan over six years, then after six years you reduce your principal to zero and your amortization period is six years.

In most relatively small purchases, the amortization term and amortization period are identical. For example, a vehicle loan has an agreed-upon interest rate and payments for a fixed term. At the end of the term, the loan is fully repaid. However, sometimes with larger purchases such as real estate transactions, financial institutions hesitate to agree to amortization terms of much more than five to seven years because of the volatility and fluctuations of interest rates. As a result, a term of five years may be established with an amortization period of 25 years. When the five years elapse, a new term is established as agreed upon between the borrower and lender. The conditions of the new term reflect prevailing interest rates and a payment plan that continues to extinguish the debt within the original amortization period.

Amortization Schedules

An amortization schedule shows the payment amount, principal component, interest component, and remaining balance for every payment in the annuity. As the title suggests, it provides a complete understanding of where the money goes, identifying how much of each payment goes to interest and how much goes to principal

An amortization schedule has five columns: 

• Payment Number. There is a row for every payment made to repay the loan.
• Payment. The periodic payment made to repay the loan. All of the payments are the same (PMT), except for the last payment.
• Interest Paid. For each row, the interest paid entry is the amount of interest due on the loan at the time of the corresponding payment.
• Principal Paid. For each row, the principal paid entry is the amount of principal repaid at the time of the corresponding payment, after the interest is paid.
• Balance. For each row, the balance records how much of the original loan amount remains after the payment is made.

To fill in an amortization schedule, you first need to have all of the details about the loan, including the loan amount (PV), the payment (PMT), the number of payments (N), and the interest rate. If any of these quantities are missing, calculate out the missing value before completing the amortization schedule.

Payment Number	Payment	Interest Paid (INT)	Principal Paid (PRN)	Balance (BAL)
0				Loan Amount1
1	PMT2	INT3	PRN4	BAL5
2	PMT2	INT3	PRN4	BAL5
N-1	PMT2
N	Final Payment10	Final INT9	Final PRN8	07
Totals	Total Amount Paid12	Total Interest Paid13	Total Principal Paid11

Follow these steps to fill in the amortization schedule.

1. In row 0, the only entry is the loan amount in the balance column.
2. Fill in the rounded loan payment down the payment column, except for the last payment.
3. Calculate the interest paid portion of the payment. The interest is the balance from the previous row times the periodic interest rate: Interest Paid = Balance from Previous Row x i.

Note: this calculation assumes that the payment frequency and the compound frequency for the interest rate are the same (i.e. a simple annuity).

4. Calculate the principal paid portion of the payment. The principal paid is the difference between the payment and the interest paid: Principal Paid = PMT - Interest Paid.

5. Calculate the outstanding balance on the loan after the payment is made. The balance is difference between the balance in the previous row and the principal paid: 

Balance = Balance from Previous Row - Principal Paid.

6. For each payment, repeat steps 2 through 5, except for the last row.

7. The last balance entry is 0. Because this is the last payment, the loan must be paid off, which means the balance is reduced to 0.
8. The final principal paid entry equals the balance entry from row N-1.
9. Calculate the final interest paid portion in the same way as in step 3.
10. The final payment is the sum of the final interest paid entry and the final principal paid entry:
Final Payment = Final INT + Final PRIN.

11. The total principal paid is the sum of the principal paid column, and is just the loan amount:
Total Principal Paid = Loan Amount.

12. The total amount paid is the sum of the payment column:
Total Amount Paid = (N-1) x PMT + Final Payment

13. The total interest paid is the sum of the interest paid column, and equals the difference between the other two column totals: Total Interest Paid = Total Amount Paid - Total Principal Paid. 

Note: The manual calculation of the interest paid entry above is based on the assumption that the payment frequency and the compounding frequency are equal. If the payment frequency and the compounding frequency are not equal, an interest conversion is required to convert the interest rate to the equivalent rate with the compounding frequency equal to the payment frequency. However, if you use the TI BAII Plus’s built-in amortization worksheet (described below), no interest conversion is required.

Example:

A $3,000 loan at 8% compounded quarterly is repaid with quarterly payments of $800. Construct the amortization schedule for the loan.

Because the payment frequency and the compounding frequency are equal, no interest conversion is required. The calculations for each entry are shown in blue. The periodic interest rate is i = 8%/4 = 2%

Business and Financial Mathematics

Corporate Finance (Book Notes)

Ch. 1 Introduction  Doc Pg 37

1.1 THE GOAL OF FINANCE: RELATIVE VALUATION

Valuation

The Law of One Price

1.2 INVESTMENTS, PROJECTS, AND FIRMS - pg. 39

Project - pg. 39

  cash flows

Firm

1.3 Firms versus Individuals

Ch. 2 The Time Value of Money and Net Present Value

2.1 OUR BASIC SCENARIO: PERFECT MARKETS, CERTAINTY, CONSTANT INTEREST RATES

The perfect market - pg. 49

2.2 LOANS AND BONDS - pg. 50

Loan 

    maturity

Bond

Interest

Fixed Income

Interest Rate

2.3 RETURNS, NET RETURNS, AND RATES OF RETURN

Return - pg. 51

    Subscript

Net return - pg. 51

Rate of return  - pg. 52

    dividends, coupons, capital gain, dividend yield, current yield, rental yield, coupon yield

Basis point - pg. 53

2.4 The Time Value of Money, Future Value, and Compounding

Time value of money

Future value pg. 54

Spot Rate pg. 57

    Connection between rate of return and FV formula - pg. 57

How Banks Quote Interest Rates - pg. 59

Annual percentage Yield (APY) 

    annual equivalent rate or effective annual rate

Interest rate

Annual Percentage Rate (APR) - pg. 60

Certificate of deposit (CD) 

2.5 PRESENT VALUES, DISCOUNTING, AND CAPITAL BUDGETING

Present Value - pg. 61

Present value formula

Discounting

Cost of capital - pg. 62

Opportunity cost

Discount factor - pg. 63

    discount rate

Repayment Schedules

Business & Financial Mathematics

Demand Loans

A demand loan is a short-term loan that generally has no specific maturity date, where the borrower can make a payment to settle the loan, either in part or full, at any time without any interest penalty, and where the lender can demand repayment in full at any time. A demand loan allows borrowing when needed and repayment when money permits, subject to the following characteristics.

1. Credit Limit. This establishes the maximum amount that can be borrowed.
2. Variable Interest Rate. Almost all demand loans use variable simple interest rates based on the prime rate. Only the best, most secure customers can receive prime, while others usually get “prime plus” some additional amount.
3. Fixed Interest Payment Date. Interest is always payable on the same date each and every month. For simplicity, the payment is usually tied to a checking or savings account, allowing the interest payment to occur automatically.
4. Interest Calculation Procedure. Interest is always calculated using a simple interest procedure based on the daily closing balance in the account. This means the first day but not the last day is counted.
5. Security. Loans can be secured or unsecured. Secured loans are those loans that are guaranteed by an asset such as a building, a vehicle, inventory, accounts receivable, etc. In the event that the loan defaults, the asset can be seized by the lender to pay the debt. Unsecured loans are those loans backed up by the general goodwill and nature of the borrower. Usually a good credit history or working relationship is needed for these types of loans. A secured loan typically enables access to a higher credit limit than an unsecured loan.
6. Repayment Structure. The repayment of the loan is either variable or fixed

• A variable repayment structure allows the borrower to repay any amount at any time, although a minimum requirement may have to be met such as “at least 2% of the current balance each month.” A current balance is the balance in an account plus any accrued interest. Accrued interest is any interest amount that has been calculated but not yet placed (charged or earned) into an account.
• A fixed repayment structure requires a fixed payment amount toward the current balance on the same date each and every month.

Some examples of demand loans include (some of these are not pure demand loans but have many characteristics of a demand loan):

1. Personal Line of Credit (LOC). A demand loan for individuals, a personal line of credit is generally unsecured and is granted to those individuals who have high credit ratings and an established relationship with a financial institution. Because a line of credit is unsecured, the credit limit is usually a small amount, such as $10,000. Repayment is variable and usually has a minimum monthly requirement based on the current balance.
2. Home Equity Line of Credit (HELOC). This is a special type of line of credit for individuals that is secured by residential homeownership. Typically, an amount not exceeding 80% of the equity in a home is used to establish the credit limit, thus enabling an individual access to a large amount of money. The interest rates tend to follow mortgage interest rates and are lower than personal lines of credit. Repayment is variable, usually involving only the accrued interest every month.
3. Operating Loans. An operating loan is the business version of a line of credit. An operating loan may or may not be secured, depending on the nature of the business and the strength of the relationship the business has with the financial institution. Repayment can be either variable or fixed.
4. Student Loans. A loan available to students to pursue educational opportunities. Although these are long-term in nature, the calculation of interest on a student loan uses simple interest techniques. These loans are not true demand loans because a student loan cannot be called in at any time. Repayment is fixed monthly.

Partial Payments of Demand Loans

The borrower can make partial payments on a demand loan at any time, without penalty, to reduce the outstanding balance on the loan. When a partial payment is made, the payment is first used to reduce the interest on the loan. If the interest is completely paid off by the payment, then the remainder of the payment is applied to reduce the principal on the loan. This approach is called the declining balance method.

NOTE

A partial payment may be more or less than the interest that is due on the loan at the time the payment is made. Each time a partial payment is made, the interest due at the time of the payment is calculated.If the partial payment is larger than the amount of interest due, then the interest is paid first and any remaining amount from the payment is used to reduce the principal.
If the partial payment is smaller than the amount of interest due, then the entire payment is applied to interest due. Any interest that is not paid-off by the payment is carried forward to the next payment. Because there is nothing leftover from the payment, nothing is applied to the principal.

References

Business and Financial Mathematics

Treasury Bills

Treasury bills, also known as T-bills, are short-term U.S. government debt obligations with maturities no longer than one year. Common maturities include 4, 8, 13, 17, 26, and 52 weeks

1. Discount Instrument: This is the most distinctive feature. Unlike bonds or notes that pay periodic interest payments (coupons), T-Bills do not pay interest directly. Instead, they are sold at a discount from their face value (or par value). When the T-Bill matures, the investor receives the full face value.

   • Example: You might buy a $1,000 T-Bill for $980.

   • When it matures, you receive $1,000.

   • Your "interest" or profit is the difference between the face value and the purchase price: $1,000 - $980 = $20.

2. T-bills do not have to be retained by the initial investor throughout their entire term. At any point during a T-bill’s term, an investor is able to sell it to another investor through secondary financial markets. Prevailing yields on T-bills at the time of sale are used to calculate the price.

Business and Financial Mathematic

Investopedia: Treasury Bills

Gemini AI 

(Can you provide a summary of how US treasury bills work and how they are similar to simple interest)

(On Bloomberg they list treasury yields with a column for price and a column for yield. For instance, 3-month, price 4.24, yield 4.34%. Can you explain these factors on a $100 par value T-Bill?)

APOLLODORUS, THE LIBRARY

 Apollodorus, The Library (also called Bibliotheca)

Book 1

Part 1. Theogony: Birth of Zeus

Primordial Beings

Origin, Gaia & Uranus, The Titan Cronus

The Birth of the Olympic Gods

Part 2. Theogony: War of the Titans

The War of the Titans

Part 3. Theogony: Olympian Gods

The Moirai - The Goddesses of Fate

Part 4. Apollo & Artemis

The Birth of Apollo & Artemis

Part 5. Demeter & Persephone

Demeter, Persephone & Hades (Pluto)

Part 6. War of the Giants, Typhon

War of the Giants, Typhon

Part 7. Prometheus, Deucalion, Daughters of Aeolus

Prometheus: The Theft of the Sacred Fire

Pandora's Box: The Story of the First Woman Created by the Gods 

The Great Flood of Greek Mythology

Hellen - The Mythological King who Gave his Name to Greece

Note 1

Part 8. Oeneus, Meleager, Tydeus

Part 9. Sons of Aeolus, Melampus, Admetus, Pelias, Argonauts

The Saga of Jason and the Argonauts

The Myth of Sisyphus - The Man Who Deceived the Gods

Book 2

Part 4 Perseus, Sons of Perseus, Amphitryon, Birth of Heracles

The Adventures of Perseus

Part 5 & 6

Hercules - The Complete Story - Greek Mythology

Book 3

Part 1 Europa, Minos, Pasiphae

Europa

Minos & Pasiphae (Second Video)

Icarus

Part 5 Dionysus, Antiope, Amphion & Zethus, Oedipus

The Story of Oedipus

Part 16. Theseus

The Origin of Theseus 1/3

The Adventures of Theseus - 2/3

Epitome (Essentially Book 3 Cont.)

Part 1. Theseus Cont.

The Adventures of Theseus - 2/3

Theseus in the Minotaur's Labyrinth - 3/3

Icarus

Part 2. Tantalus, Pelops, Atreus

Part 3. - 6. Trojan War

Part 7. Odyssey & Telegonia

Myths Not Mentioned in Apollodorus, The Library

King Midas And The Golden Touch (The Curse of Greed) - Ovid's Metamorphoses

Note 1:

Deucalion and Pyrrha had children of their own, the first of which was their son named Hellen (The ancient Greeks actually referred to themselves as Hellenes). Hellen then had three son's named Dorus, Xuthus, and Aeolus by a nymph Orseis. Each son was given a kingdom of their own. Aeolus married Enarete, daughter of Deimachus, and begat seven sons, Cretheus, Sisyphus, Athamas, Salmoneus, Deion, Magnes, Perieres, and five daughters, Canace, Alcyone, Pisidice, Calyce, Perimede

Note 2: 

Pandora only mentioned here. See Hesiod's Works and Days for story

https://www.theoi.com/Text/ApollodorusE.html

Two Variable Linear Equations: Slope of a Line

I. Intro: The Slope Of A Line

The mathematical definition of slope is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines. By just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:

First, let’s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.

Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run.

II. Finding Slope From a Graph
As we saw in the previous section, the rate of change determines whether a set of data is linear or non-linear. In this section, we formalize how to find this rate of change.

The rate of change of or the distance between the y-values compared to the x-values is called the slope. The symbol for slope is 𝑚. As we will see, graphically, the slope measures how steep the line is.

One way to determine the slope is to plot the points on a graph, draw a line through them, and then count the change in the y-values, or rise, and the change in the x-values, or run.

Rise - Vertical change between two points. 

Run - Horizontal change between two points. 

The steps to finding the slope from a graph are:

1) Select any two random points on the graph of the line (preferably with integer coordinates).
2) Label them as A and B (in any order).

3) Calculate the "rise" from A to B. While going vertically from A to B, if we have to go
"up", then the rise is positive;
"down", then the rise is negative.
4) Calculate the "run" from A to B. While going horizontally from A to B, if we have to go
"right", then the run is positive;
"left", then the run is negative.
5) Now, use the formula: slope = rise/run

Example 1

Example 2

Example 3

Khan: Intro to Slope

III. Finding The Slope From Two Points

You can find the slope of the line by counting the rise over run. As we've alluded to above, you can also find the slope of a straight line without its graph if you know the coordinates of any two points on that line. Every point has a set of coordinates: an x-value and a y-value, written as an ordered pair (x, y). The x value tells you where a point is horizontally. The y value tells you where the point is vertically.

Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates (x₁,y₁) and Point 2 has coordinates (x₂,y₂).

The rise is the vertical distance between the two points, which is the difference between their y-coordinates. That makes the rise (y₂−y₁). The run between these two points is the difference in the x-coordinates, or (x₂−x₁).

An alternate way to present the formula is:

Slope = Δy/Δx

Example

In the example you’ll see that the line has two points each indicated as an ordered pair. The point (0,2) is indicated as Point 1, and (−2,6) as Point 2. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run.

You can see from the graph that the rise going from Point 1 to Point 2 is 4, because you are moving 4 units in a positive direction (up). The run is −2−2, because you are then moving in a negative direction (left) 2 units. Using the slope formula,

Slope = rise/run = 4/-2 = -2

You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Let’s organize the information about the two points:

Can Either Point Be (X₁, Y₁)?

Let's find out.

IV: Positive & Negative Slopes

When we look at a line on a graph, its slope can fall into one of four categories. Each type of slope tells us something different about how the line behaves:

1. Positive Slope: A line with a positive slope rises as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values also increase. Think of it as going uphill.

2. Negative Slope: A line with a negative slope falls as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values decrease. Think of it as going downhill.

3. Zero Slope: A flat, horizontal line has a zero slope. It means there’s no change in y-values as the x-values increase. The line stays level.

4. Undefined Slope: A vertical line has an undefined slope. This is because the x-values don’t change, while the y-values might change infinitely. 

Khan: Positive & Negative Slope

CK-12: Introduction: Linear Equations and Inequalities of Two Variables

CK-12: Introduction to Linear Equations In Two Variables

Lumen: Slope of a Line

Cuemath: How To Find Slope From A Graph

Mathnasium: What is a Negative Slope

https://www.mathwarehouse.com/algebra/linear_equation/slope-of-a-line.php

Algebra Supporting Notes

When I started relearning pre-algebra, I used the pre-algebra section in Khan Academy as a guide. I went through the videos and found other written sources along the way. The problem with this approach is that I find Khan Academy's presentation lacking clarity. For that reason, I'm using this section to sort of reformat things by starting my conceptual understanding of the subject matter from scratch. 

I. What is Algebra?

Algebra is a branch of mathematics that deals with symbols (variables) and the rules for manipulating those symbols to represent and solve problems involving relationships between quantities. These symbols typically represent numbers, and they allow us to write mathematical expressions and equations in a generalized form. Key features include use of variables, expressions, equations, inequalities, etc. 

II. What are the different types of equations in beginning algebra?

1. Linear Equations

• Form: ax + b = 0

• Description: These equations involve variables with a power of 1 (linear) and are graphically represented as straight lines.

2. Quadratic Equations

• Form: ax² + bx + c = 0

• Description: These equations involve a variable squared ($x^2$) and are graphically represented as parabolas.

3. Absolute Value Equations

• Form: $\mathrm{\mid }ax+b\mathrm{\mid }= c$

• Description: These equations involve the absolute value of an expression, resulting in two potential solutions.

4. Rational Equations

• Form: $\frac{P(x)/}{Q(x) }= R(x)$
  

• Description: These equations involve fractions with polynomials in the numerator and/or denominator. Solutions must exclude values that make the denominator zero.

5. Radical Equations

• Form: √x + a = b

• Description: These equations include variables within a square root (or other roots). Solutions may need to be checked for extraneous results.

6. Exponential Equations

• Form: a² = b

• Description: These equations have variables as exponents. They often require logarithms to solve.

7. Inequalities (Optional in Some Courses)

• Form: $a\mathrm{x }+ \mathrm{b }> c$

• Description: These are similar to equations but involve inequality signs ($>, <, \ge , \le$).

8. Proportions

• Form: $\frac{\mathrm{a/}}{\mathrm{b }}= \frac{\mathrm{c/}}{d}$

• Description: These involve two ratios set equal to each other and are solved by cross-multiplication.

9. Systems of Equations

• Form:

  • Linear system:

    
    
    
    
    

• Description: These involve solving two or more equations simultaneously.

Understanding these types helps in identifying the approach and techniques needed to solve different algebraic problems.

III. Are linear expressions also polynomials?
Yes

A polynomial is any expression made up of variables and constants using only:

• Addition

• Subtraction

• Multiplication

• Non-negative integer exponents

A linear expression has the form:

ax+b

Where:

a and b are constants

x is the variable

The exponent on x is 1 (which is allowed in polynomials)

All linear expressions are polynomials,
but not all polynomials are linear expressions.

Two Variable Linear Equations

I. Intro

A two variables equation is any equation that includes two variables (e.g., x and y). In this section we will focus on two variable linear equations.

In linear equations with one variable the expression is compared to a fixed quantity whereas in linear equations with two variables, a relationship exists between two variables, a change in one is often followed by a change in the other.

A linear equation with two variables can be written in the form ax + by + c = 0, where a, b, c ∈ ℝ, a ≠ 0 and b ≠ 0 and x and y are variables, each raised to the first power only.
 

Here are some examples of both linear and non-linear equations.

Solutions to equations of one variable are values that make the equation true. Similarly, solutions to equations with two variables are pairs of 𝑥- and 𝑦-values that make the equation true. There are an infinite number of possible combinations of 𝑥- and 𝑦-values that will satisfy the equation.

II. Graphing a Linear Equation

The graph of a linear equation in one variable x forms a vertical line that is parallel to the y-axis and vice-versa, whereas, the graph of a linear equation in two variables x and y forms a straight line. Let us graph a linear equation in two variables with the help of the following example.

Example: Plot a graph for a linear equation in two variables, x - 2y = 2.

Let us plot the linear equation graph using the following steps.

Step 1: Convert the equation in the form of y = mx + b. This will give: y = x/2 - 1.

Step 2: Now, we can replace the value of x for different numbers and get the resulting value of y to create the coordinates. When we put x = 0 in the equation, we get y = 0/2 - 1, which results in. y = -1. Similarly, if we substitute the value of x as 2 in the equation, y = x/2 - 1, we get y = 0. If we substitute the value of x as 4, we get y = 1. The value of x = -2 gives the value of y = -2.  

Step 3: Create a table with the various coordinates.

X  |  Y

-2  |  -2

0   |  -1

2   |   0

4   |  -2

Step 4: Finally, we plot these points on a graph and draw a line through them.

III. Intercepts

The intercepts of a line are the points where the line intercepts, or crosses, the horizontal and vertical axes. To help you remember what “intercept” means, think about the word “intersect.” The two words sound alike and in this case mean the same thing.

The straight line on the graph below intercepts the two coordinate axes. The point where the line crosses the x-axis is called the x-intercept. The y-intercept is the point where the line crosses the y-axis.

x intercept: x value when y = 0

y intercept: y value when x = 0

A. Intercepts From A Graph

Finding the x and y intercepts from a graph is simply a matter of finding where the line crosses the horizontal and vertical axis. 

The x intercept is the point where the line crosses the horizontal x axis at (5, 0).

The y intercept is the point where the line crosses the vertical y axis at (0, 4).

B. Intercepts From An Equation
Notice that the y-intercept always occurs where x = 0, and the x-intercept always occurs where y = 0.

To find the x and y intercepts of a linear equation, you can substitute 0 for y and for x respectively.

For example, 3y + 2x = 6 

Solve for y = 0

3(0) + 2x = 6

2x = 6

x = 3

So the x-intercept is (3,0).

Then solve for x = 0.

3y+2(0)=6

3y = 6

y=2

The y-intercept is (0,2).

Now that you have the coordinates (3, 0) and (0, 2) you can use them to graph the line.

IV. Recognize Linear Equations

We can recognize linear equations by how the input or independent variable (𝑥 variable), and output or dependent variable (𝑦 variable) values change with respect to each other. By change, we mean what is the distance between known values. The quantity we use to compare changes is known as the rate of change. A rate is a type of ratio or fraction in which you are comparing two quantities often with different units. We see a lot of these in our daily life (60 mph, $3.99 per lb.).

Rate of Change = difference in dependent variable values/difference in independent variable values

In the following sections, rate of change will be called slope and we will find the ratio of the change in y-values to the change in x-values. 

For linear models, the rate of change is constant, that is, does not change, regardless of which data points you choose to consider. Linear relationships have a constant rate of change. If the rate of change is not constant, the relationship is not linear.

For the moment, we will consider data without units.

Example 1
Determine whether the relation in the table below is linear or non-linear.

X  |  Y
0   |  3

1   |  6

2   |  9

3   |  12

Solution: The x-values are going up by one unit at a time, while the y-values are going up by 3 units at a time. Therefore, the slope is: 3/1 = 3. Since there is a constant rate of change, the relation in the table is linear.

Example 2

Determine whether the relation in the table below is linear or non-linear.

X  |  Y

-1 |  10

0  |  7

1  |  4

2  |  1

Solution: The x-values go up by 1 unit and the y-values go down by 3 units. Therefore, the slope is: −3/1 = −3. Since there is a constant rate of change, the relation in the table is linear.


Example 3
Determine whether the relation in the table below is linear or non-linear.

X  |  Y

-2  |  5

0   |  5

2   |  5

4   |  5

Solution: The x-values are going up by 2 units at a time. On the other hand, the y-values are not going up or down, that is, the change in the y-values is 0 units. Therefore, the slope is: 0/2 = 0. Since there is a constant rate of change, the relation in the table is linear.

Example 4

Determine whether the relation in the table below is linear or non-linear.

X    |  Y

0     |  3,200

1     |  3,255

10   |  3,750

100 |  8,700

Solution: Here we are comparing the number of products produced to the costs of a business. Since the x-values do not go up by the same amount, we need to consider the ratios between individual pairs of points. The difference between 1 and 0 is 1 and the difference between their respective y-values, 3,255 and 3,200, is 55. Therefore, the slope between these two points is: 55/1 = 55.

Let's consider another pair of points. The difference between 10 and 1 is 9 and the difference between their respective y-values, 3,750 and 3,255, is 495. Therefore, the slope between these two points is: 495/ 9 = 55/1 = 55.

Another pair of values gives the difference between 100 and 10 is 90 and the difference between their respective y-values, 8,700 and 3,750 is 4,950. Therefore, the slope between these two points is: 4,950/90 = 55/1 = 55.

Note, we can choose any pair of points in the table, that is, they do not have to be consecutive. Since there is a constant rate of change, the relation is linear.

Example 5

Determine whether the relation in the table below is linear or non-linear.

X  |  Y

0   |  2

1   |  3

2   |  5

3   |  8

Solution: As the x-values go up by 1, the y-values do not increase by the same amount. First, the y-values increase by 1, then 2, then 3. Since the change in the y-values is not consistent, this relation is non-linear.

Reference

CK-12: Introduction: Linear Equations and Inequalities of Two Variables

CK-12: Introduction to Linear Equations In Two Variables

Cuemath: Linear Equations

Lumen: Graphing Linear Equations