Mathematics enthusiasts—and mathematicians themselves—have no shortage of imagination! Starting from the golden ratio, some have defined a family of numbers known as metallic means that share certain properties with the golden ratio.
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At first glance, the golden ratio, despite its mythical name, is nothing exceptional mathematically: it is simply the positive solution of a quadratic equation. Much ado about nothing? Let's see, then, what surprises it has in store.
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!
There are other ways to represent real numbers besides our decimal notation! One of the best known is to write a number as a continued fraction. The resulting representations, even when truncated, very quickly provide "good" approximations of the original number.
The Morse word, or Prouhet–Thue–Morse sequence, is an easy-to-construct mathematical object with plenty of surprises in store. Join us in exploring a combinatorial sequence that would certainly deserve to be every bit as famous as Fibonacci's!
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