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Squaring the CircleOne of the Oldest and Most Difficult Problems in Geometry
The problem of finding a square which has the exact same area as a given circle is a problem which for centuries eluded some of history's greatest mathematicians.
It was one of the greatest mathematical problems of the ancient world, and certainly one of the most deceptively difficult. The problem was simple to state, but nearly impossible (in fact, in some ways absolutely impossible) to solve: Given a circle of any size, find a square which has an area equivalent to that of the circle. Irrational Keys to SuccessWhat was the problem which so thoroughly stood in the way of this problem being solved by the earliest mathematicians? The answer is in the circumference of the circle itself, and that tricky little number: pi. Because the area of a circle cannot be found without knowledge of its circumference, and because circumference is defined by a knowledge of the value of pi (which is the ratio of a circle's circumference to its diameter, approximately 3.14), the size of any such square, in turn, would also have to be based on this number. The stumbling block in the paths of mathematicians, then, was the fact that p is fundamentally irrational. That is, while a real number, it is a number which has no end, continuing endlessly, digit after digit, with no repetition. Because a square will, by definition, depend upon the square root of this number, then, it is fundamentally impossible to find the exact dimensions of a simple square which corresponds to the area of any circle! Still, There is an AnswerDoes that mean no answer to this problem exists? Certainly not. While it may have been impossible for the ancient geometers and theorists to come up with a solution using their beloved mathematical tools (geometers like Pythagoras and Euclid prided themselves on being able to solve any geometrical problem using only a compass and a straight edge), that is not to say that no answer exists. One must simply accept the fact that the answer cannot be written in any exact form using integers. Instead, the answer must be given, of course, in terms of pi itself. Consider the simplest possible circle - the unit circle, with a radius of exactly 1 unit. The area for such a circle is given by the classic formula pi X r^2. Because r in this case equals just one, the area for a unit circle is exactly pi! Easy enough. So, to find a square with this same area becomes terribly easy: A square with the same area as a unit circle has sides of exactly the square root of pi! And that, essentially, is how a circle must be squared. More exact methods of doing this can be found using integral calculus, but the answer will always remain irrational. This is what was standing in the way of the ancient mathematicians, but which today is almost second nature for those familiar with continuous mathematics.
The copyright of the article Squaring the Circle in Math/Chaos Theory is owned by Isaac M. McPhee. Permission to republish Squaring the Circle in print or online must be granted by the author in writing.
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