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Unique Pi Value If x = (pi + 3)/2 Then 2x = pi + 3 2x(pi - 3) = (pi + 3)(pi - 3) 2(pi)x - 6x = pi^2 - 9 9 - 6x = pi^2 - 2(pi)x 9 - 6x + x^2 = pi^2 - 2(pi)x + x^2 (3 - x)^2 = (pi - x)^2 3 - x = pi - x 3 = pi But pi does not equal 3 ! What is wrong here? That the ratio of the circumference to the diameter of a circle is constant (namely, pi) has been recognized for as long as we have written records. A ratio of 3:1 appears in the following biblical verse: And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23; II Chronicles 4, 2.) The ancient Babylonians generally calculated the area of a circle by taking 3 times the square of its radius, but one Old Babylonian tablet (1900-1680 BCE) has a pi value of 3.125. The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 < pi < 22/7. Archimedes's results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. Beginning with a hexagon, he worked all the way up to a polygon with 96 sides! European mathematicians in the early modern period developed new arithmetical formulae to approximate the value of pi, such as that of James Gregory (1638-1675), which was taken up by Leibniz as : pi/4 = 1 - 1/3 + 1/5 - 1/7 . . . . The symbol for pi was introduced by the English mathematician William Jones in 1706. This symbol was adopted by Euler in 1737 and became the standard symbol for pi.



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