## Pi
Pi is not just irrational - it is also a transcendental number. This
means that not only can it not be written as a fraction, it can also not
be written as the root of any polynomial. These numbers are extremely
common (almost all real numbers are transcendental) but it can be very
difficult to pin down one number and prove that it is transcendental.
For example, root2 is irrational but not transcendental: it can be written
as the root of x Pi isn't just about circles either. Pi is one of those numbers, like
e, that crops up everywhere in mathematics. For example, in Euler's formula
in complex analysis (often known as the most beautiful mathematical formula),
pi appears once again with e and i: e Over time, people found various methods of calculating digits of pi. Archimedes found the first, which involves constructing regular polygons inside and outside a circle. Using the side lengths of the polygon to calculate the radius, he could calculate upper and lower bounds for pi by approximating the circumference by the perimeter of the polygon. He started with a hexagon, giving 3 < pi < 2root3. Eventually, by repeatedly doubling the number of sides until there were 96, Archimedes found an upper bound of 22/7. This upper bound is still used by students today as it is relatively simple but accurate - with an error under 0.05%. In fact, it is slightly closer to pi than the common decimal approximation 3.14. Another series is that 1 - 1/3 + 1/5 - 1/7... = π/4. This was found
by Leibniz in 1676 but was based on a formula for the inverse tangent
by the Indian mathematician Madhava. This unfortunately converges incredibly
slowly, and even hundreds of terms are not as good as the common approximations.
However, certain techniques can make it much faster, like Machin's formula
that π/4 = 4tan
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