Fractions

Abstract This work details a great deal of the history and manifest forms of fractions within mathematics. Rational numbers are fractions having either a terminating or repeating decimal expansion. Determining their decimal expansions, as well as the period length of repeating decimals, is completely worked out. Modern base 10 decimal expansions are compared with ancient Babylonian base 60 sexagesimal expansions. This leads to the study of infinite sums, especially to geometric series and the notions of convergence and divergence. The Fibonacci numbers are studied along with the series for 1/89 which involves them directly. The Cantor set and the Sierpinski carpet are discussed as well. Much of elementary number theory, including congruence classes, the Euler phi function, the Euclidean algorithm, and some Diophantine equations, is introduced. The mathematical properties of Farey fractions and finite continued fractions are developed. Applications of historical note include Christiaan Huygens’ cogwheeled planetarium and Archimedes’s approximation to the value of pi. The result that the rational numbers form a countable infinity is presented along with an explicit listing of all rational numbers. There is a more practical chapter concerning the fair apportionment of a cake, variations on slicing a pizza, probability questions involving markings on a stick, and ways to divide a bar of gold in order to pay wages for various numbers of days. Egyptian fractions entailing the sum of unit fractions is extensively studied. It is shown that every fraction can be written as a finite sum of distinct unit fractions. Historical examples are given along with a fractional variation on Fermat’s Last Theorem. Infinite series and Taylor series are more formally discussed in the latter chapters together with a discussion of some famous analytical formulas for pi as well as a discussion of Euler’s solution of the Basel Problem involving the sum of the reciprocals of the squares.