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 x2-2=0. The first transcendental number was constructed by Liouville but it wasn't until later that Charles Hermite proved that e was transcendental. He also said it would be extremely difficult to prove whether or not pi is irrational. This was finally accomplished by Ferdinand Lindemann.
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: eiπ+1=0. People find this beautiful as it elegantly and simply combines the five most important mathematical quantities (0, 1, e, i, and π) and the operations of addition, multiplication, and exponentiation in a single formula. Pi also shows up in probability theory. Suppose there is a floor lined with parallel lines 1unit apart. Now drop a needle of length 1 onto the floor. What is the probability that the needle intersects a line? The Comte de Buffon proved that the answer is 2/π.
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-1(1/5)-tan-1(1/239). This converges much faster and a similar formula was used to calculate a record of 620 digits by hand by Daniel Ferguson in 1946.
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