Continued Fractions
In mathematics, a continued fraction is a method of representing a fraction as the sum of an integer number and reciprocal of another number, through an iterative process. In short,
Consider the fraction 37/27. It can be written as a continued fraction as given below:
Combining all of the steps above, we can write 37/27 as a continued fraction as given below:
This can also be abbreviated by considering only the whole numbers from each step and writing them as follows: 37/27 = [1;2,1,2,3]. Similarly, 840/611 can be written as [1;2,1,2,76] (You can try it out for yourself).
Let us look at the continued fraction representation of the Golden Ratio which is quite interesting. Before that, some info about the Golden Ratio.
The Golden Ratio
The Golden Ratio is perhaps one of the most important constants in the world of mathematics. It is there in many places without us realizing it.
2 quantities(or measurements) a,b are said to be in golden ratio if their ratio is the same as the ratio of their sum to the larger among the two. To put it quite simply:
The Golden Ratio has a value approximately equal to 1.618. To see how this value is derived:
The Golden Ratio is known to appear in many buildings and artworks and also in the spiral arrangement of leaves. The Parthenon in Greece was shown to follow the golden ratio, though later studies do not support this claim.
(More details about the Golden Ratio and its relation with the Fibonacci Series will follow in a future post)
So, What Is Its Continued Fraction?
Consider the above equation in x
Substituting the value of x back in the equation, we get
Continuing in the same fashion, we get:
Writing down its abbreviated notation, we get [1;1,1,1,….]. Interesting right? Since it involves substituting the value of x back into the equation again and again, all the terms are 1.
Reforming A Prisoner?
Christopher Havens was a 40-year old in prison near Seattle for 9 years after being convicted of murder. He was sentenced to 25 years in prison and still had 16 years to go. But, it was in prison that he discovered his passion for mathematics. It was in prison that he taught himself the basics of higher mathematics.
After some time, higher mathematics was not enough to satiate his hunger for mathematics and Havens sent a letter to a mathematical publisher asking for some issues of Annals of Mathematics, a renowned journal in the field.
Mathematics As A Mission
Havens wrote that numbers had become his “mission” and that he had decided to use his time in prison for self-improvement. However, he lamented in the letter, he had no one with whom he could discuss complex mathematical topics.
An editor at Mathematica Science Publisher sent the letter on to his partner, Marta Cerruti, who forwarded the letter to her father, Umberto Cerruti, a mathematics professor from Turin. He was initially skeptical, but as a favor to his daughter he wrote an answer to Havens and gave the prisoner a problem to test his abilities.
After some time, Umberto Cerruti received the answer by mail: a 120-centimeter-long piece of paper with an incredibly long formula written on it. Cerruti first had to enter the formula in his computer to check what the prisoner had sent him: Havens had solved the tasks correctly.
The Turin mathematics professor then invited Havens to help him with an ancient math problem that Cerruti himself had been trying to solve for a long time.
Solving An Ancient Problem
Using only pen and paper, Havens tinkered for a while with the problem involving continued fractions, over which the ancient Greek mathematician Euclid had already racked his brains.
Continued fractions are not used for simple arithmetic, but to solve the approximation problems with which one approaches a result in complex calculations. Continued fractions are used in modern cryptography, which is of decisive importance today in banking and finance and in military communications.
And indeed: Havens cracked the age-old math puzzle and for the first time found some regularities in the approximation of a large class of numbers. Cerruti helped Havens to formulate the proof in a scientifically correct manner, and a few months later, in January 2020, the two of them published it in the journal Research in Number Theory.
Inspiring People
The prisoner Havens had not only managed to crack an ancient problem in mathematics. He also managed to inspire a group of fellow prisoners through his passion. A regular math club has since been formed at the prison.
Havens celebrates March 14 every year with the 14 fellow prisoners in the club as “Pi Day”. Umberto Cerruti from Turin was also took part in one of these celebrations where has was impressed by a prisoner who recited the first 461 decimal places of pi by heart, as he wrote in an article entitled “ Pi Day behind bars — Doing mathematics in prison” in the journal Math Horizons.
Havens feels doing mathematics is a way for him to pay his “debt to society,” according to Marta Cerruti, who has held several conversations with the prisoner.
(You can read the detailed article here: Murderer solves ancient math problem and finds his mission . This article was shared with me by my college junior Pranav, a few days back, which prompted me to write this post on Continued Fractions)
So, in future, if somebody tells you that mathematics has no use in real life, you can show them this article of how mathematics has helped to reform a prisoner and turn him over a new leaf.
Thanks for reading!!!
Originally published at http://infinitesimallysmallcom.wordpress.com on July 4, 2020.
