Connecting Greek Ladders and Continued Fractions - Introduction

There is a story about the followers of Pythagoras, the Pythagoreans, that seeks to explain the circumstances by which people first engaged the concept of irrational numbers. According to the story, in the 5th century B.C.E., a member of the Pythagoreans, Hippasus of Metapontum, demonstrated that the length of the hypotenuse of the right triangle with two legs of length 1 cannot be expressed as the ratio of two integers. The story goes on to say that the Pythagoreans were on a boat in the Adriatic Sea at the time of this announcement and that, in order to silence him, they threw Hippasus overboard to drown. They swore themselves to secrecy about the incident to protect the Pythagorean doctrine that “all is number” – that is, all phenomena in the universe can be reduced to whole numbers or their ratios.

Much has been written concerning this incident involving Hippasus and about the Pythagoreans in general. Some sources, such as the articles on “Pythagoras” [11] and “Pythagoreanism” [12] in the Stanford Encyclopedia of Philosophy, state that Pythagoras and his followers had very little to do with mathematics at all. It may have been that Hippasus was instrumental in a split of the Pythagoreans into two groups. One of these groups, the mathêmatici, began to focus on mathematics and studying the natural world. A few accounts credit Pythagoras himself with tossing Hippasus overboard, but this seems impossible given that history records that their lifespans did not intersect. Further, there are numerous sources that state Pythagoras himself never existed but was a legend created by the Pythagoreans. Because of their commitment to secrecy, accurate information about the life of Pythagoras and his followers is not abundant. We may never know the truth.

None of this changes the fact that at some point in human history there came the realization of quantities, such as the lengths of the diagonals of various rectangles, that could not be expressed as ratios of integers. In [4] (p. 39), Victor Katz wrote:

We do not know who discovered this result, but scholars believe that the discovery took place in approximately 430 B.C.E. And although it is frequently stated that this discovery precipitated a crisis in Greek mathematics, the only reliable evidence shows that the discovery simply opened up the possibility of some new mathematical theories.

Perhaps one of the earliest references to irrational numbers is attributed to Theodorus of Cyrene (late 5th century B.C.E.) in Plato’s Theaetetus (399 B.C.E.). From this work we learn that Theodorus demonstrated that square roots of non-square integers up to 17 are irrational. Thus, there is no solid evidence that Pythagoras (if he existed) knew about irrational numbers.

We do know for certain that significant time and effort have been spent investigating Diophantine approximations; that is, approximations of irrational numbers using rationals. The term "Diophantine" is in honor of Diophantus of Alexandria (circa 207–291 C.E.). For example, a common approximation for \(\pi\) is \(\frac{22}{7}.\) This approximation is within \(0.0013\) of the true value of \(\pi\). Further, Archimedes (circa 287 B.C.E.) offered \(\frac{265}{153}\) as an approximation for \(\sqrt{3}\) (see [10]). This approximation is within \(0.00003\) of the true value. Roger Cotes (1682–1716), an assistant of Isaac Newton, offered \(\frac{44}{37}\) as an approximation for \(\sqrt[4]{2},\) which is within \(0.00002\) of the true value (see the conclusion of [9]) .

Many techniques of Diophantine approximation have developed over the centuries. Herein we will focus on two such techniques and how they produce rational approximations for \(\sqrt{k}\) where \(k\) is a positive integer. The two techniques, continued fractions and Greek ladders, have been examined by mathematicians on many occasions (see [1], [3], [5], [6], [7], [9], and [10]). Our purpose is not to provide new insights into the efficacy of either technique but rather to demonstrate that, with a little care, the two techniques will produce the same sequence of approximations for \(\sqrt{k}\).