Tag Archives: Pi

Pi Day Countdown #14 – in search of Pi

It is unclear when the first approximations of \pi were made, approximations can be found as early as the Ancient Babylonians.  As old as \pi may be, attempts at approximating \pi is still relevant today.

The Greeks and Chinese had approximated \pi using Geometric techniques. Yet as ingenious as these techniques are, they were only able to approximate \pi with around 7 digits.

With the advent of more powerful mathematics, functions known as infinite series were created that could give the exact value of \pi, if the steps of the function were done infinitely. However, it isn’t until the invention of computing machines that the number of digits of \pi known exploded.

As of 2014, \pi has been approximated to 13 trillion digits.

Happy Pi Day

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Pi Day Countdown #13 – Pi and the Periphery

\pi is among the oldest and most famous constant of mathematics, however as a number it wasn’t called \pi until the early 18th century.

4.1.2P1
William Jones (1675 – 1749) is a Welsh Mathematician who named Pi

Before \pi was known as the number defined as the circumference of a circle divided by its diameter. However, in 1706 self taught mathematics teach William Jones labeled the constant \pi in his book “Synopsis Palmariorum Matheseos” (a new introduction to mathematics). Perhaps he named it this because \pi was Greek for p which would be first letter of the word periphery.

This notation was then popularized by the famous Swiss mathematician, Leonhard Euler in 1738, two decades after \pi‘s official naming.

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Pi day Countdown #12 – Pi and the Transcendental Number

Leonhard Euler, considered to be one of the best mathematicians of all time, had the honor of having a few Transcendental numbers named after him. The Euler’s number or 2.718281828459… is not only a beautiful and a powerful number, it also has a very interesting connection with \pi.

The Euler’s number e can be expressed in the Euler’s Identity as:

e^{ix} = \sin(x) + i \cos(x)

and with x = \pi the identity becomes

e^{i \pi} = \sin(\pi) + i \cos(\pi) = 1 or e^{i \pi} -1 = 0

Swiss Mathematician, Leonhard Euler (15 April 1707 – 18 September 1783), was famous for many discoveries in mathematics such as the Euler's number and graph theory
Swiss Mathematician, Leonhard Euler (15 April 1707 – 18 September 1783), was famous for many discoveries in mathematics such as the Euler’s number and graph theory

Which unites some of the most famous mathematical constants, not only just \pi. There is also i the number that was so controversial that it was named imaginary as a joke. The number 0 which represented the concept of nothingness, a concept that wasn’t accepted as a number for millennia. The number 1 that started it all, and not to forget \pi one of the most famous numbers in mathematics.

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Pi Day Countdown #11 – Pi and Irrationality

There is a bit of misconception that the Greeks believe that Pi was a rational number (a number that can be expressed as a fraction of two whole numbers). Many believe that it was because the Greeks did not believe in the existence of irrational numbers, yet it was the Greeks themselves who discovered and accepted the existence of Irrational numbers.

JHLambert
Johann Heinrich Lambert, 26 August 1728 – 25 September 1777, was one of the first to prove that Pi is irrational

The truth is, the Greeks approximated \pi between two fractions because that is what they knew how to do. Proof that \pi is an irrational number did not exist until the 17th century where it was proven irrational by the Swiss mathematician Johann Heinrich Lambert.

He did this by first showing that his continued fractal expansion of tan(x) holds, then he shows that if tan(x) is rational then tan(x) will have to be irrational. Since tan(\pi /4) = 1 and 1 is obviously rational, \pi/4 must but irrational and ultimately so is \pi.

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Pi Day Countdown #10 – Pi and the triangle Part 2

From the previous Pi day post we know that the function arc tangent can be used to calculate Pi, the only problem is now to approximate the value of arc tangent itself. This can be done by using a technique from calculus called Taylor series, which transforms a function into an infinite series of terms that when added produced the exact value desired.

The Taylor series of arctangent is as follows:

arctan(x) = x + x^3 / 3 - x^5 /5 + x^7/7 - x^9/9 ...

However since we know that arctan(1) = \pi /4, we only need to substitute in 1 to get a good approximation of \pi.

arctan(1) = 1 + 1 / 3 - 1 /5 + 1/7 - 1/9 ... = \pi / 4

However since it is also impossible to add up all infinite terms, the more terms that we compute this series for, the better the approximation we get for \pi / 4.

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