Explore Alan Turing's Computable Numbers & Generate Pi with Python

[fresh_42]

TL;DR

Turing’s methodology was unique: he imagined hypothetical machines that could perform complicated mathematical tasks in a deterministic manner, in the way computers do today. In this way, he inadvertently kickstarted the entire field of modern computer science…

I found this article about Alan Turing and his concept of Turing machines on the AMS website. Since we often get questions about countability and computability I thought it is worth sharing.
https://blogs.ams.org/featurecolumn/2021/12/01/alan-turing-computable-numbers/

It also contains a Python code to compute $\pi$ which by itself might be of interest:

[CODE lang="python" title="Pi"]
def generate_pi_digits():
q = 1
r = 180
t = 60
i = 2
while True:
u,y = 3*(3*i+1)*(3*i+2), (q*(27*i-12) + 5*r) // (5*t)
q,r,t,i = 10*q*i*(2*i-1), 10*u*(q*(5*i-2)+r - y*t), t*u, i+1
yield y
pi_digits = generate_pi_digits()
print(str(next(pi_digits)) + ".", end="")
for d in pi_digits: print(d,end="")
[/CODE]

 

[lofgran]

Thank you for sharing this article about Alan Turing and his concept of Turing machines. This is a very interesting topic that touches upon the foundations of computer science and mathematics.

As mentioned in the article, Alan Turing's work on computable numbers and Turing machines laid the groundwork for modern computer science and the concept of algorithmic computation. His ideas and contributions have had a significant impact on the development of technology and society as a whole.

The Python code provided in the article to compute pi is also quite fascinating. It is based on a mathematical formula that uses an infinite series of fractions to approximate the value of pi. While this may not be the most efficient method to compute pi, it is a great demonstration of the power of algorithms and how they can be used to solve complex problems.

In addition to exploring the concept of computability, Turing's work also delves into the realm of countability, which is a fundamental concept in mathematics. By understanding the limitations of what can be computed and counted, we gain a deeper understanding of the capabilities and limitations of computers and other computational devices.

Thank you again for sharing this article and the Python code. It is a great reminder of the importance of Alan Turing's contributions and their relevance to our modern world.