96 MATHEMATICS MAGAZINE Three Transcendental Numbers from the Last Non-Zero Digits of nn, Fn, and n!

In this article, we will construct three infinite decimals from the last nonzero digits of nn, Fn (the Fibonacci numbers), and n!, respectively, and we will show that all three are transcendental. Along the way, we will learn a bit about the history of transcendental numbers, discuss two major theorems in the field, and pose some questions for future research. Let’s begin by recalling what it means for a number to be transcendental. DEFINITION. For α a complex number, we say α is algebraic if it is the root of a polynomial with integer coefficients. If no such polynomial exists, we say that α is transcendental. Early mathematicians, of course, were ignorant of this distinction. Indeed, the Pythagoreans of ancient Greece believed that everything in the universe could be mea-sured by whole numbers and their ratios. It must have come as quite a shock when Hippasus (fifth century BC) first demonstrated that certain ratios, such as the ratio be-tween the diagonal and the side of a square, were not the ratios of two whole numbers. Legend has it that angry Pythagoreans threw Hippasus into the sea for his heresy [9], but the idea of irrational numbers lived on. Indeed, our word “irrational ” dates back to the Greek word αρρητoς (arrētos), meaning “unspeakable”, perhaps reflecting the Greeks ’ disgust of such messy objects as √2/1 [9], [16]. We now skip ahead to seventeenth-century Europe. With the introduction of algebra and modern notation, it finally became possible to ask if there existed numbers that were not roots of polynomials. James Gregory [3],[8] appears to have been the first to attempt to prove that pi and e were not algebraic, an ambitious (and ultimately fruitless) goal given that it was not yet known if they were even irrational (their irrationality was finally shown in the eighteenth century by Lambert and Euler, respectively). It was not until the nineteenth century that we see the first proofs of the existence of transcendental numbers. Liouville was the first to do so, with the specially-constructed numbe