The number Pi has been known for almost 4000 years. Pi is one of the most fascinating numbers.
Imagine: if you write down an alphabet and you give each letter a certain number, in some part of pi your whole future can be written. This is because pi is infinite and there is no sequence of numbers that is repeating. Pi is ratio of the circumference of a circle and its diameter. The ancient Babylonians gave very rough approximation to pi- they estimated it to 3. The first real calculations were done by Archimedes of Syracuse.
Image credit: Wikimedia Commons user John Reid
Pi is ratio of the circumference of a circle and its diameter. The ancient Babylonians gave very rough approximation to pi- they estimated it to 3. The first real calculations were done by Archimedes of Syracuse. He showed that pi is one number between $ 3 \frac{1}{4}$ and $3 \frac{10}{71}$. He approximated the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed.
Through all history, people found pi interesting. They wanted to know pi to more decimals, but couldn’t. Then the great Indian mathematical genius – Srinivasa Ramanujan started observing pi. He discovered remarkable formula for computing pi:
But it still all remained as estimation.
Later on, using the computer Alexander J. Yee and Shiger Kondo claimed to have calculated number pi to 5 trillion places. The main computation took 90 days.
Now we use the standard approximation to $\pi – 3.14$. But what if we want to remember pi to more than two decimals? We’d want to write it down as a fraction. For this we’ll use continued fraction.
Remember that the continued fractions come in form:
Let’s first take that $\pi = 3.14159$.
First number will be the floor of the starting number:
$ \mid a_0 \mid = \mid 3.14159 \mid = 3$
Now we have to calculate the first help variable $b_1$:
$ 3.14159 = a_0 + \frac{1}{b_1} \rightarrow b1 = 7.0625$
Using the help variable we can calculate $a_1$:
$ a_1 = \mid b_1 \mid = \mid 7.0625 \mid = 7 \rightarrow a_1 = 7$
Now you simply continue the process.
$ b_1 = a_1 + \frac{1}{b_2} \rightarrow b_2 = 15.96424$
$ a_2 = \mid b_2 \mid = \mid 15.9642 \mid = 15\rightarrow a_2 = 15$
$ b_2 = a_2 + \frac{1}{b_3} \rightarrow b_3 = 1.037$
$ a_3 = \mid b3 \mid = \mid 1.037 \mid = 1\rightarrow a_3 = 1$
This calculation leads us to our fraction approximation of pi:
$\pi \approx [3, 7, 15, 1] = \frac{3 + 1}{7 + \frac{1}{15 + \frac{1}{2}}}$
Now you’ll simplify this fraction and get that:
$\pi \approx \frac{335}{113} = 3.141592$
Also, don’t forget to celebrate March 14 (3/14), the international Pi Day!
