Physical objects are not in space, but these objects are spatially extended. The above excerpts — from the genius himself — precede any other person's narrative of the Theory of Relativity and the Pythagorean Theorem. Accordingly, I now provide a less demanding excerpt, albeit one that addresses the effects of the Special and General theories of relativity. The system of units in which the speed of light c is the unit of velocity allows to cast all formulas in a very simple form.
The Pythagorean Theorem graphically relates energy, momentum and mass. Euclid of Alexandria was a Greek mathematician Figure 10 , and is often referred to as the Father of Geometry.
The date and place of Euclid's birth, and the date and circumstances of his death, are unknown, but it is thought that he lived circa BCE. His work Elements , which includes books and propositions, is the most successful textbook in the history of mathematics.
In it, the principles of what is now called Euclidean Geometry were deduced from a small set of axioms. When Euclid wrote his Elements around BCE , he gave two proofs of the Pythagorean Theorem: The first, Proposition 47 of Book I, relies entirely on the area relations and is quite sophisticated; the second, Proposition 31 of Book VI, is based on the concept of proportion and is much simpler.
He may have used Book VI Proposition 31, but, if so, his proof was deficient, because the complete theory of Proportions was only developed by Eudoxus, who lived almost two centuries after Pythagoras.
Euclid's Elements furnishes the first and, later, the standard reference in geometry. It is a mathematical and geometric treatise consisting of 13 books. It comprises a collection of definitions, postulates axioms , propositions theorems and constructions and mathematical proofs of the propositions.
Euclid provided two very different proofs, stated below, of the Pythagorean Theorem. This is probably the most famous of all the proofs of the Pythagorean proposition. In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle. In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.
Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus — a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians — and others centuries after Pythagoras and even centuries after Euclid.
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making them easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics twenty-three centuries later.
Although best known for its geometric results, Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers. The geometrical system described in the Elements was long known simply as geometry , and was considered to be the only geometry possible. Today, however, this system is often referred to as Euclidean Geometry to distinguish it from other so-called Non-Euclidean geometries that mathematicians discovered in the nineteenth century.
At this point in my plotting of the year-old story of Pythagoras, I feel it is fitting to present one proof of the famous theorem.
For me, the simplest proof among the dozens of proofs that I read in preparing this article is that shown in Figure Start with four copies of the same triangle. See upper part of Figure See lower part of Figure In the seventeenth century, Pierre de Fermat — Figure 14 investigated the following problem: for which values of n are there integer solutions to the equation.
Fermat conjectured that there were no non-zero integer solutions for x and y and z when n was greater than 2.
He did not leave a proof, though. Instead, in the margin of a textbook, he wrote that he knew that this relationship was not possible, but he did not have enough room on the page to write it down. His conjecture became known as Fermat's Last Theorem. This may appear to be a simple problem on the surface, but it was not until when Andrew Wiles of Princeton University finally proved the year-old marginalized theorem, which appeared on the front page of the New York Times. Today, Fermat is thought of as a number theorist, in fact perhaps the most famous number theorist who ever lived.
It is therefore surprising to find that Fermat was a lawyer , and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous paper written as an appendix to a colleague's book.
Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books and so on with the goal of publishing his father's mathematical ideas.
Samuel found the marginal note the proof could not fit on the page in his father's copy of Diophantus's Arithmetica. In this way the famous Last Theorem came to be published. His graduate research was guided by John Coates beginning in the summer of Together they worked on the arithmetic of elliptic curves with complex multiplication using the methods of Iwasawa theory.
He further worked with Barry Mazur on the main conjecture of Iwasawa theory over Q and soon afterwards generalized this result to totally real fields. Taking approximately 7 years to complete the work, Wiles was the first person to prove Fermat's Last Theorem, earning him a place in history. Wiles was introduced to Fermat's Last Theorem at the age of He tried to prove the theorem using textbook methods and later studied the work of mathematicians who had tried to prove it.
When he began his graduate studies, he stopped trying to prove the theorem and began studying elliptic curves under the supervision of John Coates.
In the s and s, a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on some ideas that Yutaka Taniyama posed. With Weil giving conceptual evidence for it, it is sometimes called the Shimura—Taniyama—Weil conjecture. It states that every rational elliptic curve is modular. The full conjecture was proven by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor in using many of the methods that Andrew Wiles used in his published papers.
I provide the story of Pythagoras and his famous theorem by discussing the major plot points of a year-old fascinating story in the history of mathematics, worthy of recounting even for the math-phobic reader. It is more than a math story, as it tells a history of two great civilizations of antiquity rising to prominence years ago, along with historic and legendary characters, who not only define the period, but whose life stories individually are quite engaging.
Greek mathematician Euclid, referred to as the Father of Geometry, lived during the period of time about BCE , when he was most active. His work Elements is the most successful textbook in the history of mathematics. Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus, a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians centuries after Pythagoras and even centuries after Euclid.
There is concrete not Portland cement, but a clay tablet evidence that indisputably indicates that the Pythagorean Theorem was discovered and proven by Babylonian mathematicians years before Pythagoras was born. So many people, young and old, famous and not famous, have touched the Pythagorean Theorem. The eccentric mathematics teacher Elisha Scott Loomis spent a lifetime collecting all known proofs and writing them up in The Pythagorean Proposition, a compendium of proofs.
The manuscript was published in , and a revised, second edition appeared in Surprisingly, geometricians often find it quite difficult to determine whether some proofs are in fact distinct proofs.
In addition, many people's lives have been touched by the Pythagorean Theorem. A year-old Albert Einstein was touched by the earthbound spirit of the Pythagorean Theorem. The wunderkind provided a proof that was notable for its elegance and simplicity. That Einstein used Pythagorean Theorem for his Relativity would be enough to show Pythagorean Theorem's value, or importance to the world. But, people continued to find value in the Pythagorean Theorem, namely, Wiles.
The theorem's spirit also visited another youngster, a year-old British Andrew Wiles, and returned two decades later to an unknown Professor Wiles. Young Wiles tried to prove the theorem using textbook methods, and later studied the work of mathematicians who had tried to prove it. When he began his graduate studies, he stopped trying to prove the theorem and began studying elliptic curves, which provided the path for proving Fermat's Theorem, the news of which made to the front page of the New York Times in Sir Andrew Wiles will forever be famous for his generalized version of the Pythagoras Theorem.
How far away from the base of the shed does he need to place the ladder? Draw in the right angled triangle and add the lengths given:. We know the hypotenuse so we will need to subtract :. Round to two decimal places.
The bottom of the ladder needs to be 2. Angharad is buying a new 32 inch screen TV. Remember that TV screens are measured along the diagonal.
If it is 18 inches tall, how wide is the screen? Her TV cabinet is 28 inches wide. Makes sense, right? The smaller triangles were cut from the big one, so the areas must add up. And the kicker: because the triangles are similar, they have the same area equation. Let's call the long side c 5 , the middle side b 4 , and the small side a 3. Our area equation for these triangles is:.
Now let's play with the equation:. This takes a bit of time to see, but I hope the result is clear. How could the small triangles not add to the larger one?
Actually, it turns out the Pythagorean Theorem depends on the assumptions of Euclidean geometry and doesn't work on spheres or globes, for example. But we'll save that discussion for another time.
We used triangles in our diagram, the simplest 2-D shape. But the line segment can belong to any shape. Take circles, for example:. Pretty wild, eh? We can multiply the Pythagorean Theorem by our area factor pi, in this case and come up with a relationship for any shape.
Remember, the line segment can be any portion of the shape. We could have picked the circle's radius, diameter, or circumference -- there would be a different area factor, but the relationship would still hold. So, whether you're adding up pizzas or Richard Nixon masks, the Pythagorean theorem helps you relate the areas of any similar shapes. Now that's something they didn't teach you in grade school. The Pythagorean Theorem applies to any equation that has a square.
In reality, the "length" of a side can be distance, energy, work, time, or even people in a social network:. Metcalfe's Law if you believe it says the value of a network is about n 2 the number of relationships. In terms of value,. Pretty amazing -- the 2nd and 3rd networks have 70M people total, but they aren't a coherent whole. The network with 50 million people is as valuable as the others combined.
Some programs with n inputs take n 2 time to run bubble sort, for example. In terms of processing time:. Pretty interesting.
Given this relationship, it makes sense to partition elements into separate groups and then sort the subgroups. Indeed, that's the approach used in quicksort, one of the best general-purpose sorting methods.
The Pythagorean theorem helps show how sorting 50 combined elements can be as slow as sorting 30 and 40 separate ones. We don't often have spheres lying around, but boat hulls may have the same relationship they're like deformed spheres, right? Assuming the boats are similarly shaped, the paint needed to coat one 50 foot yacht could instead paint a 40 and footer.
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