You might be required in social studies to choose a person who has had a huge impact on many cultures and many people throughout time. It could be that for a science project you investigate the life of a scientist, such as an astronomer, who made important discoveries about what we know today about our earth or other planets. Or perhaps the music teacher will set you an assignment to discover a character who made a musical breakthrough.
For any one of these assignments you could select the same person to study, Pythagoras. Usually, when we hear the name Pythagoras, or more formally Pythagoras of Samos, we think of right-angled triangles or the hypotenuse and maybe squares and things! But there was a lot more to Pythagoras than his famous theorem.
He was well known in his day, enough that statues of him were sculptured and drawings and paintings made. If you look for information about him at a maths history site, like the wonderfully informative one produced by St Andrew's University , you will discover that after Nash, Einstein and Newton, Pythagoras' is the most requested biography of a mathematician.
He lived from about BC to about BC in Greece, we can't be sure exactly, but it was a long time ago and we are as fascinated by him today as people in his time were. Little is known of Pythagoras' childhood. The only description of how he looked that is probably true is the description of a noticeable birthmark on his thigh! Information differs, some sources say that he had two brothers, although others state it was three. What they agree on is that he was well educated; he was a fine musician, he played the lyre and used music to help people who were ill; he learned poetry and was able to recite famous and popular Greek writers like Homer.
While he was a young student, three teachers who were philosophers greatly influenced Pythagoras. When he was betwee n 18 and 20 years old, Pythagoras left Greece and went to a town called Miletus, which is in the country we now call Turkey, and visited an old man named Thales. Thales made a big impression on him and advised Pythagoras to travel to Egypt.
While he was there he visited many temples and took part in discussions with the priests and learned from some of Thales' pupils about geometry and cosmology. Because of the people he met and the experiences he had, Pythagoras became a philosopher like his teachers, but went on to make important discoveries in mathematics, astronomy, and the theory of music. Pythagoras was taken prisoner and taken to Babylon where he continued his quest for learning new things.
He was instructed in the sacred rites of the Bab ylonians and learnt about their mystical worship of the gods. He perfected his skills in music and arithmetic as well as the other mathematical sciences taught by the Babylonians.
Although he was a very important figure in the development of mathematical ideas, we don't know much about Pythagoras' actual mathematical achievements. Unlike many other Greek mathematicians, none of his writings exist to provide evidence about his interests. Fortunately, many other people wrote about him and his work and from them we know that Pythagoras was really into numbers in a big way!
In fact, he thought numbers had personalities - he regarded each one as either masculine or feminine, perfect or incomplete, beautiful or ugly. This number interest probably came about after his capture by the Babylonians. It's what we now call an irrational number, not because it is illogical, but because it can't be represented as a ratio of whole numbers. This was what sent the Pythagoreans into such a spin that they may have sacrificed poor Hippasus.
If you believe that everything is constructed from whole numbers, it is a terrible a shock to discover that there is an everyday number, a 'real world' number like the diagonal of a square, that doesn't fit your picture of the world. It's a nightmare - and one from which the Pythagoreans would never really recover. And that makes top an even number too - because an odd number multiplied by itself is always odd.
But hang on. We started out by saying top and bottom were chosen as the simplest possible ratio - any common factors had already been divided out. Now we are saying they have to have a common factor of 2. This isn't possible. As Hippasus discovered to his cost, that inscription over the Pythagorean school All is number would have to be extended to cope with more complex ideas than ratios of whole numbers. It is based on a slightly different proof that the square root of 2 is irrational.
This alternative proof can be generalised to prove that all square roots of whole numbers that are not square numbers are irrational, that is the square roots of 3, 5, 6, 7, 8, 10, To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. Register for our mailing list. University of Cambridge. 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.
Maor, E. Google Scholar. Leonardo da Vinci 15 April — 2 May was an Italian polymath someone who is very knowledgeable , being a scientist, mathematician, engineer, inventor, anatomist, painter, sculptor, architect, botanist, musician and writer. Leonardo has often been described as the archetype of the Renaissance man, a man whose unquenchable curiosity was equaled only by his powers of invention.
He is widely considered to be one of the greatest painters of all time and perhaps the most diversely talented person ever to have lived.
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