According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen.
The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers.
They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea. The Pythagorean Theorem is a statement about triangles containing a right angle.
The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides. Figure 1. According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C. Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras.
It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem. Therefore, the square on c is equal to the sum of the squares on a and b.
Burton There are many other proofs of the Pythagorean Theorem. One came from the contemporary Chinese civilization found in the oldest extant Chinese text containing formal mathematical theories, the Arithmetic Classic of the Gnoman and the Circular Paths of Heaven. The proof of the Pythagorean Theorem that was inspired by a figure in this book was included in the book Vijaganita, Root Calculations , by the Hindu mathematician Bhaskara.
Bhaskara's only explanation of his proof was, simply, "Behold". These proofs and the geometrical discovery surrounding the Pythagorean Theorem led to one of the earliest problems in the theory of numbers known as the Pythgorean problem. Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:.
The formula that will generate all Pythagorean triples first appeared in Book X of Euclid's Elements :. In his book Arithmetica , Diophantus confirmed that he could get right triangles using this formula although he arrived at it under a different line of reasoning.
The Pythagorean Theorem can be introduced to students during the middle school years. This theorem becomes increasingly important during the high school years. It is not enough to merely state the algebraic formula for the Pythagorean Theorem.
Students need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials. Through the use of manipulatives and other educational resources, the Pythagorean Theorem can mean much more to students than just. Indian mathematicians in the ancient times knew the Pythagorean theorem, they also used something called the Sulbasutras of which the earliest date from ceremonial axe ca.
Apart from India, the Chinese and the Egyptians also used this theorem in construction. This is how many of the Egyptians pyramids are built. The Egyptians wanted a perfect degree angle to build the pyramids which were actually two right-angle triangle whose hypotenuse forms the edges of the pyramids. There are some clues that the Chinese had also developed the Pythagoras theorem using the areas of the sides long before Pythagoras himself.
But they did not actually write them down and so Pythagoras gets the credit for simply writing them down. Pythagoras was born in around BC, in an island called Samos in Greece. There is no much information about his youth though he did a lot of travelling to study is all that is known. Latter Pythagoras settled in Crotone a city and comune in Calabria , where he started his cult or group called the Pythagoreans. The Pythagoreans loved maths so much that it was like a god to them, they pretty much worshiped maths.
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. Loomis, E. A rational number is a number that can be expressed as a fraction or ratio rational.
The numerator and the denominator of the fraction are both integers. When the fraction is divided out, it becomes a terminating or repeating decimal. The repeating decimal portion may be one number or a billion numbers. Rational numbers can be ordered on a number line.
An irrational number cannot be expressed as a fraction. Irrational numbers cannot be represented as terminating or repeating decimals. Irrational numbers are non-terminating, non-repeating decimals. Schilpp, P. Okun, L. Physics-Uspekhi Article Google Scholar. Download references. You can also search for this author in PubMed Google Scholar.
Also, check out few more interesting articles related to Pythagoras Theorem for better understanding. Example 1: Consider a right-angled triangle. The measure of its hypotenuse is 16 units. One of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula?
Example 2: Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long.
How far should she place the base of the ladder away from the building? We can visualize this scenario as a right triangle. We need to find the base of the right triangle formed. Example 3: Kate, Jack, and Noah were having a party at Kate's house. After the party gets over, both went back to their respective houses. Jack's house was 8 miles straight towards the east, from Kate's house. Noah's house was 6 miles straight south from Kate's house. How far away were their houses Jack's and Noah's?
We can visualize this scenario as a right-angled triangle. That means Jack and Noah are hypotenuses apart from each other. The converse of Pythagoras theorem is: If the sum of the squares of any two sides of a triangle is equal to the square to the third largest side, then it is said to be a right-angled triangle.
The Pythagoras theorem works only for right-angled triangles. When any two values are known, we can apply the Pythagoras theorem and calculate the other. The square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides.
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