The Great Mathematical Problems
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Search: Search. The Great Mathematical Problems. Add to Basket. Add to Wishlist Add to Compare. About the book There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics. He is a Fellow of the Royal Society, appears frequently on radio and television, and does research on pattern formation and network dynamics.
Reviews Stewart's imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating It will enchant math enthusiasts as well as general readers who pay close attention Britain's most brilliant and prolific populariser of mathematics Praise for previous books: 'This is not pure maths. Thoroughly entertaining Stewart has served up the instructive equivalent of a Michelin-starred tasting menu, or perhaps a smorgasbord of appetisers. Close It appears you don't have the ability to view PDFs in this browser.
Click here to download the sample directly. Also by this author Do Dice Play God? Ian Stewart Multiple formats. Significant Figures Ian Stewart Multiple formats. Calculating the Cosmos Ian Stewart Multiple formats. The War On Heresy R. Another impossible problem in Euclidean geometry is squaring the circle.
Squaring the circle means constructing a square whose area is the same as a circle, again only by compass and straightedge. The trick here is squaring the circle by any method is impossible.
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The proof for constructing regular polygons and the disproof for squaring the circle are not found in Euclidean geometry but in analytic geometry, requiring algebra, calculus and complex number theory. The disproof of squaring the circle comes from the irrational nature of pi. Irrational means that pi cannot be represented exactly by a fraction. Pi is not only irrational but also transcendental; it cannot be represented by algebraic expression.
The two problems above provide the author the means of telling a story and also provide the reader a feel for how difficult mathematics problems get solved.
Over a period of years, sometimes centuries, various people prove a conjecture to be true but only with special restrictions. With each improvement, the proof gains by relaxing the restrictions. When a breakthrough is made, a simpler and cleverer idea either completes the proof or provides a new starting point for further advances. The first open problem offered up by Stewart is the Goldbach conjecture, formulated years ago, and involves prime numbers and efficient methods of factorialization.
The conjecture can be solved by computer but computer generated solutions are inefficient; the effort of working through the computations takes more time as the problem scales—in fact the universe will end before a computer finds or does not find a solution. Little progress has been made—there are more efficient methods, but these methods are probabilistic, that is, they do complete but sometimes fail to provide an answer. Another difficult problem that was eventually solved is the four-color theorem.
This theorem, with no real use outside of pure mathematics is simple to describe. If one were to color a map and adjacent areas had to be given different colors, then only four colors would be needed. This holds true for any map; the difficulty is in finding the proof. In , Appel-Haken provided a proof of the four-color theorem, which consisted of pages of calculations.
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Given the complexity of the solution, how do you know the proof is correct? Proofs are valued for their mathematical beauty and simplicity and this proof was not beautiful. This was proof by exhaustion, something that addressed every possible case, which was unsatisfying to mathematicians. A beautiful, simple proof is still in the waiting. Another seemingly impossible problem came out of a digression.
Inside was a brief remark, a puzzle that would not be solved for years. The Kepler conjecture considers the most efficient way to pack spheres. Imagine a holiday gift box of oranges—successive layers of oranges fitting into the gaps of previous layers. As with the four-color theorem, computers were needed to generate the proof.
The Mordell conjecture is another problem notorious for the ease of its stating compared to the difficulty in finding a proof. Belonging to an area of number theory called Diophantine equations, solutions have to be rational, whole, and positive. Readers will be entertained with the incredible effort it took to construct the proof.
The three-body problem is not only an open problem of interest to mathematicians but also to astronomers as it reflects gravitational orbits, based on physical theory. Quantum theory is overextended for one body, and quantum field theory runs into trouble with no bodies—vacuum.
The three-body is another problem that can be stated simply. Mathematicians in the late s were unable to solve the three-body problem, which by reached some pages of differential equations. Computers were first used for the three-body problem in the late s but it was discovered that the calculations converged so slowly as to not be practical. This spurred others to seek solutions that converged more rapidly. If the solution to the three-body problem were impossible to solve then more complex orbits would be even more so.
Take for example, our solar system.