# 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.

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 $e^{i \pi} -1 = 0$

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.

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 #4 – Pi and Archimedes

The in third century AD, Archimedes of Syracuse devised a simple, yet ingenious way of approximating Pi.

First what he did was draw a circle, and then inside the circle he would inscribe a regular polygon, such as a hexagon, so that each corner of the shape touched the circle. Next he drew another regular polygon, lets say a hexagon again, outside of the circle so that the midpoint of the edges touch the circle.

By doing so Archimedes can accurately measure the perimeters of the inner and outer polygons. From this he knew that the circumference of the circle cannot be larger then the perimeter of the outer polygon. At the same time the circumference has to be larger then the perimeter of the inner polygon. By dividing the perimeters by (2 $\times$ radius), Archimedes has successful trapped the value of $\pi$ between two values.

Thus by using regular polygons with more and more sides, Archimedes manages to find that $\pi$ was somewhere between $22/7$ and $223/71$.

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# Pi Day Countdown

H! This is WuFeng and I would like to talk about Pi. Pi, 3.14159… is one of the oldest and most famous mathematical constants known to humanity. It is simple in definition yet has so many intriguing facts, details and applications.

Pi has many applications in computing, engineering and science. It raises many questions and facts in mathematics and computer science. This number has been a part of human scientific and engineering achievement since the early days of civilization as there are evidence of this number even during Babylonian times.

For us this number had inspired the Famous Numbers column as well as an article called “Importance of a circle“.

For two weeks starting on March 1, 2015 to March 14, we will be posting each day one new post about Pi as a count down to Pi day on 3/14/15.

With best Wishes

WuFeng

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