![]() In short, the initial square of side m was not the smallest possible one that satisfies the geometric equation. That means each side of the new small squares is equal to the integer n – m, and each side of the large purple square is equal to the integer n – 2( n – m) = 2 m – n. In other words, there are two equal and smaller squares ( red, B) that together have the same area as the larger square ( purple, B). Consequently, the area not covered by the two blue squares is equal to twice the area covered by both of them. Our reasoning requires that the area of the two blue squares must be equal to the area of the red one ( A). The two red squares in that shape must have the same area as the central purple square that is created where the two blue squares overlap. Wedging the two blue squares into two diagonally opposite corners of the red square produces a new shape. The assumption would be valid only if there is no smaller positive integer that satisfies it. We assume that in our drawing, m is the smallest positive integer satisfying this equation. If so, there exists a square, with sides equal to n, whose area equals twice that of a square with sides equal to m. Say that √2 is the quotient of n and m-that is, 2 = n 2/ m 2, or 2 m 2 = n 2. It also shows that it is wrong to believe that everything simple has already been discovered: brilliant yet astonishingly simple ideas are still waiting to be revealed. Because even if he did not create it, the proof offers a perfect example of Conway’s approach to mathematics, which he demonstrated in 100 different ways. You might ask whether it was Conway himself who formulated the proof. He attributed the creation of the proof to mathematician Stanley Tennenbaum, who, according to Conway, had abandoned mathematics. Though there are many proofs of this theorem, the most intuitive is a very simple little drawing that Conway included in a lecture published in a book in 2005. ![]() This first negative finding in mathematics showed that humans do not create the laws governing numbers but rather uncover them as they explore uncharted mathematical territory. The discovery and its proof were profoundly unsettling for mathematicians. The discovery of the irrationality of √2 is credited to Pythagoras or one of his disciples, although we do not know whether the reasoning behind it was arithmetic or geometric. It cannot be expressed by the quotient of positive integers n and m, or n/m. One of the most surprising and important mathematical findings is the irrationality of the square root of 2 (√2)-the length of the diagonal of a square with sides that are one unit long. He was able to invent objects and problems in many different domains and to solve the most recalcitrant puzzles, conceiving methods no one could have imagined. But it would take several volumes to do justice to this exceptional mind. I am going to cover several different topics to which Conway contributed. The best way to honor him, it seems to me, is to highlight the kind of mathematics that amazed and fascinated him. ![]() Conway liked simple, practical mathematical problems that called on his creativity. In preparing the articles, I would often come across a result he had demonstrated or an important idea that he had been the first to propose on the topic, much to my surprise. Conway’s commitment to mathematics for everyone led him to work on puzzles that delight fans of recreational mathematics, such as the famous Collatz conjecture (which is discussed toward the end of this article).Ĭonway was the mathematician most cited (more then 30 times!) in my columns in Pour la Science, the French edition of Scientific American. People who worked with Conway report that he thought so fast that no sooner did he hear a problem stated than he often already had a solution. This class comprises integers, real numbers, transfinite numbers and infinitesimals-a structure that no one previously imagined was possible in which everything can be added, multiplied, and so on. Conway also devised elegant puzzles for packing boxes of blocks that can only be solved efficiently with clever reasoning.Īccording to Conway, his most important contribution was his conceptualization of a marvelous system of numbers called surreal numbers. The areas of research covered by this remarkable mathematician included group theory, node theory, geometry, analysis, combinatorial game theory, algebra, algorithmics and even theoretical physics.Ĭonway’s inclinations and talent led him to invent a remarkable cellular automaton called the Game of Life, which continues to fascinate after 50 years. On April 11, 2020, John Horton Conway died of COVID-19 at the age of 82 in New Brunswick, N.J.
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