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Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

32 + 42 = 52

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=2 is a right triangle, but (1,1,2) is not a Pythagorean triple because 2 is not an integer. Moreover, 1and 2 do not have an integer common multiple because 2 is irrational.

Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.

Plimpton 322, a Babylonian clay tablet from about 1800 BC
Plimpton 322, a Babylonian clay tablet from about 1800 BC

One example of a Pythagorean triple is a=3, b=4, and c=5: Ancient Egyptians used this group of Pythagorean triples to measure out right angles. They would tie knots in a piece of rope to create 3, 4, and 5 equal spaces. Three people would then hold each corner of the rope and form a right triangle!

3-4-5 triangle using rope in Egypt
3-4-5 triangle using rope in Egypt

right triangles during the construction process to help determine the slope of the pyramid. The Pythagorean Theorem states that given a right triangle with sides of length a and b respectively and a hypothenuse of length c, the lengths satisfy the equation a2 + b2 = c2.

Pyramids at Giza
All three of Giza’s famed pyramids and their elaborate burial complexes were built during a frenetic period of construction, from roughly 2550 to 2490 B.C.

When searching for integer solutions, the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b.

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3 things I learned while my plane crashed

Ric Elias had a front-row seat on Flight 1549, the plane that crash-landed in the Hudson River in New York in January 2009. What went through his mind as the doomed plane went down? At TED, he tells his story publicly for the first time.

Ric Elias is the CEO and cofounder of Red Ventures, a portfolio of fast-growing digital businesses.

Why you should listen

Ric Elias was given the gift of a miracle: to face near-certain death, and then to come back and live differently.

Video 4m 45s

Ric Elias – Ted Talks

A native of Puerto Rico, Elias attended Boston College and Harvard Business School before starting his career as part of GE’s Financial Management program. He cofounded Red Ventures in 2000, just months before the dot-com bubble burst. The company weathered the storm; by 2007 it was ranked fourth on the Inc. 500 list, and in 2015 the company was valuated at more than $1 billion. Elias has cultivated an award-winning company culture, ranking as a “Best Place to Work” in Charlotte, North Carolina, for ten years in a row.

Elias’s leadership style and personal life are deeply influenced by his experience as a survivor of Flight 1549, also known as the “Miracle on the Hudson.” He is devoted to using his platform to “leave the woodpile higher than he found it” — spinning out multiple nonprofits from Red Ventures over the years, all of which are aimed at creating educational opportunity and economic mobility for under-served groups. In 2018, Elias launched Forward787, a social enterprise committed to raising and deploying $100 million to build businesses in Puerto Rico that compete with the world’s top companies. In 2019, he launched a podcast, 3 Things with Ric Elias, as a continuation of the learning journey he shared on the TED stage.

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Michio Kaku: 3 mind-blowing predictions about the future

What lies in store for humanity? Theoretical physicist Michio Kaku explains how different life will be for your descendants—and maybe your future self, if the timing works out.

15 min

with
Michio Kaku

Michio Kaku: 3 mind-blowing predictions about the future
  1. We will become a space-faring species
  2. We will expand the brain’s capabilities
  3. We will defeat cancer

About

Michio Kaku (Japanese: カク ミチオ, 加来 道雄, born January 24, 1947) is an American theoretical physicist, futurist, and popularizer of science (science communicator). He is a professor of theoretical physics in the City College of New York and CUNY Graduate Center. Kaku is the author of several books about physics and related topics and has made frequent appearances on radio, television, and film. He is also a regular contributor to his own blog, as well as other popular media outlets. For his efforts to bridge science and science fiction, he is a 2021 Sir Arthur Clarke Lifetime Achievement Awardee.

Sources
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Warren Buffet | 3 Life Decisions

Warren Buffett Says 3 Decisions in Life Separate
High Achievers From Mere Dreamers. If the third-richest man in the universe says it, who’s to argue?

BY MARCEL SCHWANTES, INC. CONTRIBUTING
EDITOR AND FOUNDER, LEADERSHIP FROM THE CORE. 7/14/2022

Warren Buffet​
Warren Buffet

Warren Buffett is smarter than me. Much smarter. But, outside of his complete mastery of all things related to investment, is the earthly wisdom Buffett imparts on us mind-boggling? Probably not. It’s the fact Warren Buffett is saying it. He is articulating, in the simplest of terms, things our eighth-grade teacher could have told us, but their coming from Buffett is what makes all the difference.

Nearing the age of 90, the Oracle of Omaha is a success juggernaut whose common sense resonates deep within our souls. Some of his advice just might transform you, but you need to apply it. Here are three inspiring Buffett lessons to move you from dreamer to high achiever.

  1. Don’t risk what you have to get something you don’t need. Buffett once advised graduating students at the University of Florida that he has witnessed both businesses and individuals put themselves at risk to chase after bigger things, usually out of greed when they should have held back.

Buffett said, “If you risk something that is important to you for something that is unimportant to you, it just doesn’t make sense. I don’t care if the odds you succeed are 99 to 1 or 1,000 to 1.”

  1. Invest in relationships with honest and ethical people.

He also asked University of Florida students to think of a classmate they felt had the makings of success long term, such that they would want to get 10 percent of that person’s earnings for the rest of their lives.
“You would probably pick the one you responded the best to, the one who has the leadership qualities, the one who is able to get other people to carry out their interests,” said Buffett. “That would be the person who is generous, honest, and who gave credit to other people for their own ideas.”

  1. Measure your life’s success through one word: love.
    In the Buffett biography, The Snowball: Warren Buffett and the Business of Life, Buffett explains that the highest measure of success in life comes “by how many of the people you want to have love you actually do love you.”

Some people die filthy rich and get buildings named after them but “the truth is that nobody in the world loves them,” says Buffett. In the end, the ultimate test of how you’ve lived your life comes down to love.
“The trouble with love is that you can’t buy it. You can buy sex. You can buy testimonial dinners. But the only way to get love is to be lovable. You’d like to think you could write a check: I’ll buy a million dollars’ worth of love. But it doesn’t work that way. The more you give love away, the more you get,” asserts Buffett.

Warren Buffet​
Warren Buffet

Source: Inc.

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The Metre – the repeating circle & triangulation

The Metre (meaning measure) was one ten-millionth of the distance from the North Pole to the Equator! France embarked on a first large scale measurement. It took 7 years to measure the distance from Dunkirk to Barsalona. They used triangulation with an instrument called the Repeating Circle along with trigonometry.

The standardization of measurement: the Metre

Creating the Metre – a universal standard

By the 16th century, there we over 250,000 weights and measures in Europe. This effected trade, navigation, building plans, etc. Fire hoses would not connect from town to town. France chose to create a standard by measuring something unchangeable. They chose the Earth. Before this standardization, the human body (the Ruler of the land) would make new measurements upon gaining power.

The Repeating Circle

Repeating Circle
Repeating Circle

DESCRIPTION

This is one of two double repeating circles that Ferdinand Rudolph Hassler, the first superintendent of the U. S. Coast Survey, ordered from Edward Troughton in London in 1812, and that was shipped in 1815. The large circle may be angled from vertical to horizontal to the opposite vertical position. It is graduated to 10 minutes, and read by four verniers and two magnifiers to single minutes.

A repeating circle is a geodetic instrument with two telescopes that is designed to reduce errors by repeated observations taken on all parts of the circumference of a circle. The form was developed by the Chevalier de Borda, first executed by Etienne Lenoir in Paris around 1789, and popular for about 50 years.

Ref: F. R. Hassler, “Papers on Various Subjects Connected with the Survey of the Coast of the United States,” Transactions of the American Philosophical Society 2 (1825): 232-420, on 315-320 and pl. VII. “The Repeating Circle Without Reflection, as made by Troughton,” in The Cyclopaedia: or, Universal Dictionary of Arts, Sciences, and Literature, edited by Abraham Rees (London, 1819), Vol. VII, Art “Circle.”

Image credit:

NAME: repeating circle MAKER: Troughton and Simms PLACE MADE: United Kingdom: England, London MEASUREMENTS: overall: 32 1/8 in x 26 3/4 in x 17 in; 81.6356 cm x 67.945 cm x 43.18 cm upper circle: 17 1/2 in; 44.45 cm circle at base: 13 1/2 in; 34.29 cm telescope: 24 in; 60.96 cm overall; base: 16 3/4 in x 15 1/4 in x 16 in; 42.545 cm x 38.735 cm x 40.64 cm overall; horizontal circle: 13 in x 23 in x 20 in; 33.02 cm x 58.42 cm x 50.8 cm ID NUMBER PH.314640 CATALOG NUMBER 314640 ACCESSION NUMBER 208213


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Trilogies as Persuasion | 1 min watch

Little Lawyer Lesson #1: Use Trilogies #shorts. Robert Gouveia Esq.

Trilogies can be very powerful tools of persuasion.
Terence McCarthy, author of MacCarthy on
Cross-Examination, explains just how important
trilogies are in trials.

Little Lawyer Lesson #1: Use Trilogies

Robert Gouveia (formerly Robert Gruler) is a
criminal defense lawyer in Scottsdale, Arizona,
and host of Watching the Watchers, a show
focused on Accountability, Transparency and
Justice.

Source: https://youtube.com/shorts/2Pg82ju0J3c?feature=share

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Security & Exchange Commission

March 6, 2022.

Gary Gensler (SEC Chairman) interview on Jon Stewart speaking about the three pillars of the SEC mission.

  • Investor protection
  • Facilitating capital formation
  • Facilitating fair orderly markets
Jon Stewart interviewing Gary Gensler (SEC)

Gary Gensler is an American government official and former investment banker serving as the chair of the U.S. Securities and Exchange Commission. Gensler previously led the Biden–Harris transition’s Federal Reserve, Banking, and Securities Regulators agency review team.

Jon Stewart is an American comedian, writer, producer, director, political commentator, actor, and television host. He hosted The Daily Show, a satirical news program on Comedy Central, from 1999 to 2015. Stewart now hosts The Problem with Jon Stewart, which premiered September 2021 on Apple TV+. Wikipedia

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3 Self-Help Books to Read If You Want To Change Your Life

Scottish author Samuel Smiles unknowingly gave birth to a new sub-genre of non-fiction with his 1859 work, Self-Help. In it, Smiles offered inspiring quotes for working class individuals who wanted to write their own rags-to-riches story through perseverance. The book sold over a quarter of a million copies, making it a bestselling title of its time. Medium reported that it also marked the beginnings of the self-help book genre, which is popular for a reason: it aims to provide insight and actionable advice for its target readers. And there are many who simply seek wisdom and comfort in what they read.

If you’re on a similar quest for self-improvement, these three titles might give you what you’re looking for:

Quiet: The Power of Introverts in a World That Can’t Stop Talking by Susan Cain

According to writer Susan Cain, nearly a third of the population classify as introverts and many more have introverted qualities. Her book, Quiet, gave many readers power during the lockdowns of COVID-19 as social needs became difficult to fulfil in isolation. It’s a comforting read that teaches us that many of the influential figures in history, from Rosa Parks to Tom Hanks, are introverts themselves. They’ve had great success in a society that very much favors the “extrovert ideal”.

A Business Insider book review highlights three key takeaways for readers:

  1. Society still values extroversion over introversion in business, politics, education, and other spheres — and this needs to change.

  2. Introverts and extroverts can create a dynamic relationship as they’ll have more success collaborating.

  3. There’s nothing wrong with being an introvert (or having introverted traits) and it’s time to find your power in it. If you want to know how, Cain’s Quiet is a great place to start.

The Power of Habit: Why We Do What We Do in Life and Business by Charles Duhigg

Journalist Charles Duhigg’s bestselling book The Power of Habit: Why We Do What We Do in Life and Business breaks down the science behind our habits and rituals and how we can change our detrimental ones. He argues that these habits — whether it’s on exercise, productivity or financial success — are within our control. That said, our post on The Power of Habits notes that the process of reforming habits contains three important elements: a cue, a routine, and a reward. When you are able to recognize these three, you can change any habit and turn it into one that benefits you. So whether you’re trying to, say, drop your smoking habit or be less of a micromanager to your employees, The Power of Habit is a helpful how-to guide for self-improvement.

Why We Sleep: Unlocking the Power of Sleep and Dreams by Matthew Walker

Sleep is, neuroscientist Matthew Walker’s book argues, the most important human need, yet it is also one of the most overlooked. The book explains the science behind how sleep works, but it is also a deep dive into why we need to at all. He also brings to light the health challenges that you can experience with lack of sleep, such as lowered immune function, reduced cognitive skills, and repressed growth. Getting quality sleep (not just sleeping more) can truly change your life in at least three ways:

  1. It can make you less vulnerable to disease, more focused, and more energetic.

  2. Sleep can regulate your mood and reduce your risk of mental health conditions like anxiety and depression.

  3. The healthier you are physically and mentally, the more energy and time you can devote to the important aspects of your life, such as your work, hobbies, and relationships.

If you’re having trouble hitting the hay, take a crack at Matthew walker’s self-help book. He himself says in the introduction, “So please, feel free to ebb and flow into and out of consciousness during this entire book. I will take absolutely no offense. On the contrary, I would be delighted.”

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Bill Maher | the easiest three predictions in the world

Some presidents spend their post presidency building homes for the poor, or raising money for charity, or painting their toes. Trump has spent his figuring out how to pull off the coup he couldn’t pull off last time. Here’s the easiest three predictions in the world.

* Trump will run in 2024

* He will get the Republican nomination

* And whatever happens on election night the day he will announce that he won

I’ve be saying it ever since he lost.

Bill Maher | Real Time | October 8 2021
Bill Maher | Real Time
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Pawn Stars | Silver Dollar Certificate

This series opens the doors to the only family-run pawnshop in Las Vegas, where three generations of the Harrison family use their sharp-eyed skills to assess what’s real and what’s fake. Objects the colourful customers bring in range from the obscure to the truly historic, and it’s up to the guys at the Gold & Silver Pawn Shop – with help at times from their network of experts – to reveal the sometimes surprising answer to `What’s this worth?’.

Genre: Reality television. Original release: July 19, 2009 – present
Pawn Stars | Silver Dollar Certificate

Rick talks about three different people making the engraving for the silver dollar certificate to prevent counterfeiting.

Rick Harrison
Pawn Stars shop cast
Pawn Stars shop cast | Rick Harrison, Chumlee, Corey Harrison & Richard Harrison
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After spending 5 years studying millionaires, I’ve found that there are 3 types of people who end up the wealthiest

Thomas C. Corley, Contributor Sep 8, 2018, 10:18 PM

• To become wealthy there are a few things you have to do with your money. 

• Thomas C. Corley studied and interviewed 233 wealthy individuals over the course of four years and found three common ways they build their fortunes.

• “Saver-investors” focused on having no debt, lived well below their means, and invested and saved for many years. 

• “Virtuosos” were the best of the best in their careers, and worked for companies that gave stock options or owned their own highly-profitable businesses. 

• “Dreamers” were the wealthiest group: They pursued a big dream and made it a reality, which led to some massive gain or income.

What I found is that there were three predominant paths rich people pursued in order to accumulate their wealth.

1. The ‘saver-investors’

Just less than 22% of the rich people in my Rich Habits Study fell into this category. The “saver-investors” all had zero debt, and the passive income generated by their invested savings was enough to meet or exceed their standard of living.

They all had five things in common:

1. They had a low standard of living and

2. They typically made a modest income and

3. Their modest income exceeded their low standard of living and

4. They saved 20% or more of their modest income for many years and

5. They consistently and prudently invested their savings for many years.

It took the Savers about 32 years to accumulate an average wealth of $3.4 million.

2. The ‘virtuosos’

Approximately 27% of the rich people in my study were “virtuosos.” These rich people were virtuosos in their career, industry, or profession. They were among the best at what they did.

These individuals either worked for large, publicly-held corporations, in which a significant portion of their compensation was stock-based compensation or they were entrepreneurs/small business owners with enterprises that were highly profitable.

It took the virtuosos about 20 years to accumulate an average wealth of $4 million.

3. The ‘dreamers’

The “dreamers” were by far the wealthiest group in my study. Approximately 51% of them were individuals who pursued some big dream and were able to turn that dream into a reality. Their dream eventually provided them with an enormous amount of income, profit, or gain, and they accumulated an average of $7.4 million in about twelve years.

The point to all of this is: There is more than one way to skin a cat. If you’re risk averse, it does not disqualify you from becoming rich. If you have no dream or you’re not interested in saving your way to wealth, becoming a virtuoso in what you do for a living can make you rich. If you are not a saver or a virtuoso, pursuing some dream that makes your heart sing can also make you wealthy.

If you want to be rich, the only important thing is to pick one path that works for you and stick with it for many years. The one common denominator all levels of wealth shared was time — it took many years to accumulate their wealth.

Source: https://www.businessinsider.com/3-ways-to-build-wealth-2018-7

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Richard Branson’s 3 Best Tips for Overcoming Self-Doubt

Richard Branson

Richard Branson

The Virgin founder and all-around adventurer’s best tricks for pushing past self-doubt.

As Richard Branson’s recent trip to space proves, the Virgin founder and all-around adventurer isn’t the kind of guy to let doubt stand in the way of an outlandish goal. But just because Branson personifies the swashbuckling, risk-taking entrepreneur, that doesn’t mean he never experiences self-doubt. 

Like his fellow space company founder Elon Musk, Branson confessed recently on LinkedIn that he, too, is human and is sometimes haunted by doubts. He just refuses to be stopped by them. 

So how does he overcome fear when it threatens to stand in the way of his outsize dreams? His first move, he writes, is to remind himself “of all the brilliant innovations and discoveries that would have come to nothing if their inventors had given in to their doubts.” His next is to try a handful of practice fear-busting techniques he graciously shares with readers. 

1. Talk it out.

Left in your head, doubts can metastasize or return in an endless loop of unproductive rumination. Rather than worry over the same fear over and over again, it’s better to let the air in with a little conversation and a fresh perspective.  

“It’s always a good idea to discuss any doubt you have with colleagues and friends, and to really listen to their feedback,” Branson advises. “If you feel more confident after these conversations, take a ‘screw it, let’s do it’ attitude and push your doubts to the side.” 

2. Examine your doubt.  

And what if your doubt lingers? Then maybe it’s time to listen to it. Sometimes doubt is there for a reason, Branson acknowledges, and the best way to deal with it is to actually solve whatever issue is causing you to worry. 

“Your doubts may signal a niggling problem that needs to be addressed. If you demand proof from your doubt, you’ll be able to either squash it or solve the problem. Either way, your dreams and ideas will be better for it,” Branson writes, recalling a time in the early days of Virgin Atlantic when a moment of doubt led to a business-saving course correction. 

3. Take a break. 

Like any good Brit, Branson’s final suggestion starts off with a suggestion to “get a cup of tea.” But don’t worry if you’re not a fan of hot steeped beverages. The point isn’t the cuppa, it’s the idea that sometimes the best solution for negative thoughts and tough obstacles is a little distance. 

“Whenever doubt starts to get the better of me, I find exercise really helps. I’ll get on my bike, play some tennis, or have some downtime with my family. This is often when my best ideas come to me as well!” Branson continues. 

He’s far from the only achiever to notice the magic powers of a change of scene, a little exercise, and some time out of doors. A host of thinkers from Charles Darwin to Steve Jobs to Albert Einstein have recommended a little hiking, sailing, or bike riding whenever you’re hung up on a problem or worry. 

Science shows nature reduces stress, exercise makes us smarter, and rhythmic activities that demand some but not all of our attention — like walking or shampooing your hair — can help shake out creative ideas. No wonder taking a break can actually be the fastest, best way to work your way around self-doubt. 

Source: www.inc.com/jessica-stillman/richard-branson-self-doubt-fear-confidence.html