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Theorem
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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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Pythagorean Theorem for Babies
Set the children in your life on a lifelong path to learning with the next installment of the Baby University board book series.Full of scientific information, this is the perfect book to teach complex concepts in a simple, engaging way.Pythagorean Theorem for Babies is a colorfully simple introduction for youngsters (and grownups!) to what the Pythagorean Theorem is and how we can go about proving it.It's never too early to become a scientist!
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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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Would fire-breathing dragons be biologically possible?
No, fire-breathing dragons would not be biologically possible. The ability to breathe fire would require a biological mechanism to produce and expel a flammable substance, as well as a way to ignite it. There are no known biological systems that could accomplish this, and the energy required to produce and expel fire would be impractical for a living organism. Additionally, the heat and pressure from breathing fire would likely be harmful to the dragon's own body. Therefore, fire-breathing dragons are purely a product of mythology and fiction.
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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
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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.
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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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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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Dragons : Meet the Legendary Monsters of Mythology
An epic guide to the history and mythology of dragons from around the world for fantasy-mad kids by medieval historian Dr Cait Stevenson. For thousands of years humans have feared or revered dragons, be they winged fire-breathing monsters from Europe or slithering water gods from Asia.In this book, featuring beautiful illustrations courtesy of Cinthya Alvarez, readers will be charmed by ancient myths and learn about the cultures that gave birth to these legendary monsters.Includes the stories of: Fáfnir, a dragon from Norse mythology whose greed gets the better of him The female dragon-slayer from the Arabic folktales One Thousand and One Nights Apophis, the dragon god of ancient Egyptian myth who stalks the underworldReaders will also learn about the link between dragons and dinosaurs, meet real-life dragons such as Komodo dragons and bearded dragons, study dragon constellations, encounter the dragons of J.R. R. Tolkien’s Middle Earth, and wonder at the popularity of games such as Dungeons & Dragons. It’s everything kids who love dragons could ever want to know!
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Notes on the Brown-Douglas-Fillmore Theorem
Suitable for both postgraduate students and researchers in the field of operator theory, this book is an excellent resource providing the complete proof of the Brown-Douglas-Fillmore theorem.The book starts with a rapid introduction to the standard preparatory material in basic operator theory taught at the first year graduate level course.To quickly get to the main points of the proof of the theorem, several topics that aid in the understanding of the proof are included in the appendices.These topics serve the purpose of providing familiarity with a large variety of tools used in the proof and adds to the flexibility of reading them independently.
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Birth of a Theorem : A Mathematical Adventure
“This man could plainly do for mathematics what Brian Cox has done for physics” — Sunday TimesHow does a genius see the world?Where and how does inspiration strike?Cédric Villani takes us on a mesmerising adventure as he wrestles with the Boltzmann equation – a new theorem that will eventually win him the most coveted prize in mathematics and a place in the mathematical history books.Along the way he encounters obstacles and setbacks, losses of faith and even brushes with madness. His story is one of courage and partnership, doubt and anxiety, elation and despair.Of ordinary family life blurring with the abstract world of mathematical physics, of theories and equations that haunt your dreams and seeking the elusive inspiration found only in a locked, darkened room. Blending science with history, biography with myth, Villani conjures up an inimitable cast: the omnipresent Einstein, mad genius Kurt Godel, and Villani’s personal hero, John Nash. Step inside the magical world of Cédric Villani…
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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.
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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.
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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.
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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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