# π

• Historically [analytic geometry] arose... from the comparison of curvilinear and rectilinear magnitudes. ...the Egyptians and Babylonians, in their geometry of the circle, took the first steps. The former made a remarkably accurate estimate of the ratio of the area of the circle to the area of the square on the diameter, taking the ratio to be ${\displaystyle (1-{\frac {1}{9}})^{2}}$, equivalent to taking a value of about 3.16 for ${\displaystyle \pi }$. The Babylonians adopted the cruder approximation 3... (although an instance is known in which the value is taken as ${\displaystyle 3{\frac {1}{8}}}$), but... recognized that the angle inscribed in a semicircle is right, anticipating Thales by well over a thousand years. Moreover, they were familiar... with the Pythagorean theorem.
• One of the most frequently mentioned equations was Euler's equation, ${\displaystyle e^{i\pi }+1=0.\,\!}$ Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.
${\displaystyle {\frac {4}{\pi }}={\frac {3}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {5}{6}}\cdot {\frac {7}{6}}\cdot {\frac {7}{8}}\cdots }$