Continued fractions—stacks of fractions that never end—have been studied by some of the greatest mathematicians, including Euler and Lagrange, as well as Galois. Let's take a closer look at these fractions and their strange denominators.
Articles recommended for you.
Real numbers can be written without making an arbitrary choice of base. Euclid's algorithm provides a method that extends to all real numbers and leads to the notion of a continued fraction. In this notation, the golden ratio becomes the simplest irrational number to write!
Since π is defined as the ratio of a circle's circumference to its diameter, it would be convenient if it were rational—that is, the quotient of two integers. Unfortunately, it is not! Worse still, some might say, it is even transcendental…
Among irrational numbers, transcendental numbers are not roots of any polynomial equation with integer coefficients. They seem to transcend common sense, hence their name. Cantor showed that they make up the overwhelming majority of the real numbers, without identifying a single one!
Adding several fractions always produces another fraction. But when the sum continues indefinitely, things are very different. If chosen carefully, such sums can approximate numbers like π and yield a wealth of fascinating, unexpected results.
Discussion
Sign in to post a comment and talk with other readers.
No comments yet. Be the first to respond.
