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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.

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.

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!

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.

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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Alan Alda | The 3 Rules of 3

We Were Built to Connect with Other People — Here’s How to Be Better At It. Before you follow another “tip” or “trick,” there’s something Alan Alda wants you to know.

His best tip to become a better communicator is what he calls the three rules of three. Listen to his practical hints for becoming a communication pro but, as he remarks, try to get there organically through the process. Alan Alda’s most recent book is If I Understood You, Would I Have This Look on My Face?

“So the first rule is, I try only to say three important things when I talk to people”.

“The second rule is, if I have a difficult thing to understand, if there’s something I think is not going to be that easy to get, I try to say it in three different ways”.

“And the third tip, which I always forget, is that if I have a difficult thing that’s hard to get, I try to say it three times through the talk”.

—- Alan Alda

Alan Alda doesn’t want you to take “pro tips” from anyone-not even Alan Alda. When it comes to his area of expertise public speaking and empathetic communication there are no hacks or shortcuts; if you want to be a world class public speaker, you have to earn those stripes through the process of deeply understanding what it is to talk, listen, and connect.

Alda calls tips intellectual abstractions; it’s akin to the difference between information and knowledge, between parroting a few words in French and speaking the actual language. But, when pushed by yours truly at Big Think, Alda does give up the goods (willingly we promise no Alan Aldas were harmed in the making of this video).

5 min Video

Alan Alda has earned international recognition as an actor, writer and director. In addition to The Aviator, for which he was nominated for an Academy Award, Alda’s films include Crimes and Misdemeanours, Everyone Says I Love You, Flirting With Disaster, Manhattan Murder Mystery, And The Band Played On, Same Time, Next Year and California Suite, as well as The Seduction of Joe Tynan, which he wrote, and The Four Seasons, Sweet Liberty, A New Life and Betsy’s Wedding, all of which he wrote and directed. Recently, his film appearances have included Tower Heist, Wanderlust, and Steven Spielberg’s Bridge of Spies.

Source: https://bigthink.com/neuropsych/alan-alda-we-were-built-to-connect-with-other-people-heres-how-to-be-better-at-it/

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Noise by Daniel Kahneman | 3 Distinctions

The Michael Shermer Show with Daniel Kahneman – Noise: A Flaw in Human Judgment

DESCRIPTION

Imagine that two doctors in the same city give
different diagnoses to identical patients. Now
imagine that the same doctor making a different
decision depending on whether it is morning or
afternoon, or Monday rather than Wednesday.
This is an example of noise: variability in
judgments that should be identical.

Shermer speaks with Nobel Prize winning
psychologist and economist Daniel Kahneman
about the detrimental effects of noise and what
we can do to reduce both noise and bias, and
make better decisions in: medicine, law, economic
forecasting, forensic science, bail, child
protection, strategy, performance reviews, and
personnel selection.

Video clip – 3 minutes

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