Squaring the Circle
Date: 12/22/97 at 12:19:20 From: deloach Subject: Squaring the circle I understand that the problem of squaring the circle implies that, using only a compass and straightedge, you can't construct a square with the same area as a circle with a given radius. My question / unease is hard to express, but it's something like: can you determine such a square at all? How can you determine precisely the square root (the side of the square) of a number that has been arrived at from multiplication with a number such as pi, a number that never repeats the decimal? Aren't you just determining a number that's as accurate as you wish, but not THE answer?
Date: 12/22/97 at 14:11:27 From: Doctor Rob Subject: Re: Squaring the circle Equivalently, you cannot construct a line segment whose length is the perimeter of a circle whose radius is given. If you could, you could then construct a line segment whose length was Sqrt[Pi] times the radius, and that would be the length of the side of the square in question. You can't determine this square root exactly, just as you cannot determine the square root of 2 exactly, since it, too, is an irrational number You can, however, construct a line segment whose length is Sqrt times the length of a given line segment. Sqrt and Sqrt[Pi] are two valid real numbers. In a sense, I think you are asking about irrational real numbers, or perhaps even transcendental real numbers. How do we specify one of them if we cannot write its full decimal expansion? For algebraic ones, they are specified by the polynomial equation they solve. Sqrt is the positive root of x^2 - 2 = 0, for example. Often we don't worry about the actual value of Sqrt, but leave it as a symbol, and whenever we see Sqrt^2, replace it with 2. The same trick works with the imaginary unit i, where i^2 + 1 = 0. For transcendental ones, we often give an infinite series, an infinite product, or an infinite sequence which converges to the number, or a definite integral whose value is the number. We also often just give it a symbol, and work with the symbol, until we *really* need to know the numerical value to some degree of accuracy. For example, Pi = 4*[1 - 1/2 + 1/3 - 1/4 + 1/5 - ...], Pi = 4*(1-1/3^2)*(1-1/5^2)*(1-1/7^2)*..., 1 Pi = 4*Integral 1/(1+x^2) dx, 0 e = 1 + 1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*5) + ..., e = lim (1 + 1/n)^n. n->infinity Then, at least in theory, one can compute the number in question to as much accuracy as desired. For Sqrt[Pi], there is the following infinite product: Sqrt[Pi] = Sqrt*(2/3)*(4/5)*(6/7)*(8/9)*... To actually compute Sqrt[Pi], one would undoubtedly compute Pi to many digits, then take its square root by one of the standard methods. This is because the above expressions are too slowly convergent to be useful for computations of extreme accuracy, but there are very fast methods to compute Pi as the limit of very rapidly convergent infinite sequence. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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