Simple proofs: The impossibility of trisection

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Introduction: Ancient Greek mathematicians developed the methodology of “ruler-and-compass” constructions: if one is given only a ruler (without marks) and a compass, what objects can be constructed as a result of a finite set of operations? While they achieved many successes, three problems confounded their efforts: (1) squaring the circle; (2) trisecting an angle; and (3) duplicating a cube (i.e., constructing a cube whose volume is twice that of a given cube). Indeed, countless mathematicians through the ages have attempted to solve these problems, and countless incorrect “proofs” have been

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Simple proofs: The fundamental theorem of algebra

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The fundamental theorem of algebra is the assertion that every polynomial with real or complex coefficients has at least one complex root. An immediate extension of this result is that every polynomial of degree $n$ with real or complex coefficients has exactly $n$ complex roots, when counting individually any repeated roots.

This theorem has a long, tortuous history. In 1608, Peter Roth wrote that a polynomial equation of degree $n$ with real coefficients may have $n$ solutions, but offered no proof. Leibniz and Nikolaus Bernoulli both asserted that quartic polynomials of

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Simple proofs: The irrationality of pi

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Mankind has been fascinated with $\pi$, the ratio between the circumference of a circle and its diameter, for at least 2500 years. Ancient Hebrews used the approximation 3 (see 1 Kings 7:23 and 2 Chron. 4:2). Babylonians used the approximation 3 1/8. Archimedes, in the first rigorous analysis of $\pi$, proved that 3 10/71 < $\pi$ < 3 1/7, by means of a sequence of inscribed and circumscribed triangles. Later scholars in India (where decimal arithmetic was first developed, at least by 300 CE), China and the Middle East computed $\pi$

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Simple proofs of great theorems

Euler’s identity Credit: Redbubble.com

Mathematics and beauty

Modern mathematics is one of the most enduring edifices created by humankind, a magnificent form of art and science that all too few have the opportunity of appreciating. The great British mathematician G.H. Hardy wrote, “Beauty is the first test; there is no permanent place in the world for ugly mathematics.” Mathematician-philosopher Bertrand Russell added: “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music,

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New books and articles on the “great silence”

Credit: NASA

The great silence

As we have explained in previous Math Scholar blogs (see, for example, MS1 and MS2), the perplexing question why the heavens are silent even though, from all evidence, the universe is teeming with potentially habitable exoplanets, continues to perplex and fascinate scientists. It is one of the most significant questions of modern science, with connections to mathematics, physics, astronomy, cosmology, biology and philosophy.

In spite of the glib dismissals that are often presented in public venues and (quite sadly) in writings by some professional scientists (see MS1 and MS2 for examples and rejoinders), there

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