Three Transcendental Numbers from the Last Non-Zero Digits ofnn,Fn, andn!

In this article, we will construct three infinite decimals from the last nonzero digits of n n , Fn (the Fibonacci numbers), andn!, 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 ! a complex number, we say ! is algebraic if it is the root of a polynomial with integer coefficients. If no such polynomial exists, we say that ! 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 measured 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 between 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 !#$ o% (arr¯