The Joy of x: A Guided Tour of Math, from One to Infinity

The great innovation here is that even though this system is based on the number 10, there is no single symbol reserved for 10. Ten is marked by a position—the tens place—instead of a symbol. The same is true for 100, or 1,000, or any other power of 10. Their distinguished status is signified not by a symbol but by a parking spot, a reserved piece of real estate. Location, location, location.

As the numbers inside the logarithms grew multiplicatively, increasing tenfold each time from 100 to 1,000 to 10,000, their logarithms grew additively, increasing from 2 to 3 to 4. Our brains perform a similar trick when we listen to music. The frequencies of the notes in a scale—do,re, mi, fa, sol, la, ti, do—sound to us like they’re rising in equal steps. But objectively their vibrational frequencies are rising by equal multiples. We perceive pitch logarithmically.

The analysts of the 1800s identified the underlying mathematical cause of the Gibbs phenomenon. For functions (or, nowadays, images) displaying sharp edges or other mild types of jump discontinuities, the partial sums of the sine waves were proven to converge pointwise but not uniformly to the original function. Pointwise convergence means that at any particular point x, the partial sums get arbitrarily close to the original function as more terms are added. So in that sense, the series does converge, as one would hope. The catch is that some points are much more finicky than others. The Gibbs phenomenon occurs near the worst of those points—the edges in the original function.

In the early part of the ninth century, Muhammad ibn Mus al-Khwarizmi, a mathematician working in Baghdad, wrote a seminal textbook in which he highlighted the usefulness of restoring a quantity being subtracted by adding it to the other side of the equation. He called this process al-jabr (Arabic for “restoring”), which later morphed into “algebra.” Then, long after his death, he hit the etymological jackpot again. His own name, al-khwarizmi, lives on today in the word “algorithm.”