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  • Theorem
    Theorem


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  • Pythagorean Theorem
    Pythagorean Theorem


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  • Fermat’s Last Theorem
    Fermat’s Last Theorem

    ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.’ It was with these words, written in the 1630s, that Pierre de Fermat intrigued and infuriated the mathematics community.For over 350 years, proving Fermat’s Last Theorem was the most notorious unsolved mathematical problem, a puzzle whose basics most children could grasp but whose solution eluded the greatest minds in the world.In 1993, after years of secret toil, Englishman Andrew Wiles announced to an astounded audience that he had cracked Fermat’s Last Theorem.He had no idea of the nightmare that lay ahead. In ‘Fermat’s Last Theorem’ Simon Singh has crafted a remarkable tale of intellectual endeavour spanning three centuries, and a moving testament to the obsession, sacrifice and extraordinary determination of Andrew Wiles: one man against all the odds.

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  • Fermat’s Last Theorem
    Fermat’s Last Theorem

    Introducing the Collins Modern Classics, a series featuring some of the most significant books of recent times, books that shed light on the human experience – classics which will endure for generations to come. ‘Maths is one of the purest forms of thought, and to outsiders mathematicians may seem almost otherworldly’ In 1963, schoolboy Andrew Wiles stumbled across the world’s greatest mathematical problem: Fermat’s Last Theorem.Unsolved for over 300 years, he dreamed of cracking it. Combining thrilling storytelling with a fascinating history of scientific discovery, Simon Singh uncovers how an Englishman, after years of secret toil, finally solved mathematics’ most challenging problem. Fermat’s Last Theorem is remarkable story of human endeavour, obsession and intellectual brilliance, sealing its reputation as a classic of popular science writing. ‘To read it is to realise that there is a world of beauty and intellectual challenge that is denied to 99.9 per cent of us who are not high-level mathematicians’ The Times

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  • What is the Pythagorean theorem and the cathetus theorem?

    The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, it can be written as a^2 + b^2 = c^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides, called catheti. The cathetus theorem, also known as the converse of the Pythagorean theorem, states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right-angled triangle. In other words, if a^2 + b^2 = c^2, then the triangle is a right-angled triangle, where c is the longest side (hypotenuse) and a and b are

  • What is the Pythagorean theorem and the altitude theorem?

    The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as a^2 + b^2 = c^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. The altitude theorem, also known as the geometric mean theorem, states that in a right-angled triangle, the altitude (the perpendicular line from the right angle to the hypotenuse) is the geometric mean between the two segments of the hypotenuse. This can be expressed as h^2 = p * q, where h is the length of the altitude, and p and q are the lengths of the two segments of the hypotenuse.

  • How could one structure a research paper on the Pythagorean theorem?

    A research paper on the Pythagorean theorem could be structured in the following way: 1. Introduction: Provide an overview of the Pythagorean theorem, its history, and significance in mathematics. 2. Historical background: Discuss the origins of the theorem, its discovery by Pythagoras, and its development over time. 3. Mathematical explanation: Explain the theorem itself, including the formula (a^2 + b^2 = c^2) and how it is used to calculate the length of the hypotenuse in a right-angled triangle. 4. Applications: Explore real-world applications of the Pythagorean theorem in various fields such as architecture, engineering, and physics. 5. Conclusion: Summarize the key points discussed in the paper and highlight the importance of the Pythagorean theorem in mathematics and beyond.

  • How can the altitude theorem and the cathetus theorem be transformed?

    The altitude theorem and the cathetus theorem can be transformed by applying them in different geometric shapes and contexts. For example, the altitude theorem, which states that the length of the altitude of a triangle is inversely proportional to the length of the corresponding base, can be applied to various types of triangles and even extended to other polygons. Similarly, the cathetus theorem, which relates the lengths of the two perpendicular sides of a right triangle to the length of the hypotenuse, can be generalized to other right-angled shapes or even applied in three-dimensional geometry. By exploring different scenarios and shapes, these theorems can be adapted and transformed to solve a wide range of geometric problems.

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  • Emmy Noether's Wonderful Theorem
    Emmy Noether's Wonderful Theorem

    "In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began."-Albert Einstein The year was 1915, and the young mathematician Emmy Noether had just settled into Gottingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity.Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy.Knowing of her expertise in invariance theory, they requested Noether's help.To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetries-one of the most important pieces of mathematical reasoning ever developed. Noether's "first" and "second" theorem was published in 1918.The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity.The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems.General relativity, it turns out, exhibits local gauge invariance.Noether's theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions.In Dwight E. Neuenschwander's new edition of Emmy Noether's Wonderful Theorem, readers will encounter an updated explanation of Noether's "first" theorem.The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the "second" theorem, including Noether's resolution of concerns about general relativity.Other refinements in the new edition include an enlarged biography of Emmy Noether's life and work, parallels drawn between the present approach and Noether's original 1918 paper, and a summary of the logic behind Noether's theorem.

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  • Godel's Theorem : A Very Short Introduction
    Godel's Theorem : A Very Short Introduction

    Very Short Introductions: Brilliant, Sharp, Inspiring Kurt Gödel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago.The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions.Gödel's Theorem has since established itself as a landmark intellectual achievement, having a profound impact on today's mathematical ideas.Gödel and his theorem have attracted something of a cult following, though his theorem is often misunderstood. This Very Short Introduction places the theorem in its intellectual and historical context, and explains the key concepts as well as common misunderstandings of what it actually states.A. W. Moore provides a clear statement of the theorem, presenting two proofs, each of which has something distinctive to teach about its content.Moore also discusses the most important philosophical implications of the theorem.In particular, Moore addresses the famous question of whether the theorem shows the human mind to have mathematical powers beyond those of any possible computerABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area.These pocket-sized books are the perfect way to get ahead in a new subject quickly.Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

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  • Lectures on the h-Cobordism Theorem
    Lectures on the h-Cobordism Theorem

    Important lectures on differential topology by acclaimed mathematician John MilnorThese are notes from lectures that John Milnor delivered as a seminar on differential topology in 1963 at Princeton University.These lectures give a new proof of the h-cobordism theorem that is different from the original proof presented by Stephen Smale.Milnor's goal was to provide a fully rigorous proof in terms of Morse functions.This book remains an important resource in the application of Morse theory.

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  • Lectures on the h-Cobordism Theorem
    Lectures on the h-Cobordism Theorem

    Important lectures on differential topology by acclaimed mathematician John MilnorThese are notes from lectures that John Milnor delivered as a seminar on differential topology in 1963 at Princeton University.These lectures give a new proof of the h-cobordism theorem that is different from the original proof presented by Stephen Smale.Milnor's goal was to provide a fully rigorous proof in terms of Morse functions.This book remains an important resource in the application of Morse theory.

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  • What are the altitude theorem and the cathetus theorem of Euclid?

    The altitude theorem of Euclid states that in a right-angled triangle, the square of the length of the altitude drawn to the hypotenuse is equal to the product of the lengths of the two segments of the hypotenuse. This theorem is also known as the geometric mean theorem. The cathetus theorem of Euclid states that in a right-angled triangle, the square of the length of one of the catheti (the sides that form the right angle) is equal to the product of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that cathetus. This theorem is also known as the Pythagorean theorem. Both the altitude theorem and the cathetus theorem are fundamental principles in the study of geometry and are essential for understanding the properties of right-angled triangles.

  • What is Thales' theorem?

    Thales' theorem states that if A, B, and C are points on a circle where the line AC is a diameter, then the angle at B is a right angle. In other words, if a triangle is inscribed in a circle with one of its sides being the diameter of the circle, then that triangle is a right triangle. Thales' theorem is a fundamental result in geometry and is named after the ancient Greek mathematician Thales of Miletus.

  • What is the difference between similarity theorem 1 and similarity theorem 2?

    Similarity theorem 1, also known as the Angle-Angle (AA) similarity theorem, states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. On the other hand, similarity theorem 2, also known as the Side-Angle-Side (SAS) similarity theorem, states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. The main difference between the two theorems is the criteria for establishing similarity - AA theorem focuses on angle congruence, while SAS theorem focuses on both side proportionality and angle congruence.

  • What is the proof for the altitude theorem and the cathetus theorem?

    The altitude theorem states that in a right triangle, the altitude drawn from the right angle to the hypotenuse creates two similar triangles with the original triangle. This can be proven using the properties of similar triangles and the Pythagorean theorem. The cathetus theorem states that the two legs of a right triangle are proportional to the segments of the hypotenuse that they create when an altitude is drawn from the right angle. This can also be proven using the properties of similar triangles and the Pythagorean theorem.

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