Why do we count in tens? Probably because we have ten fingers. There is nothing special about the number 10: any number can be written in any base. Leibniz took this idea to the extreme and asked: what is the smallest number of digits that can write everything? The answer is two: 0 and 1. Each place is worth double The numbers 0 to 32 written in binary, in the table from the paper Explanation of Binary Arithmetic, which Leibniz published in 1703 in the memoirs of the Paris Academy of Sciences.G. W. Leibniz (1703) · Wikimedia Commons · public domain In the decimal system the places are worth 1, 10, 100, 1000… and each digit says how many times we use that value. In the binary system the places are worth 1, 2, 4, 8, 16, 32… and each digit can only say yes (1) or no (0).
Historical portrait The portrait of Leibniz painted by Christoph Bernhard Francke around 1695 comes from Wikimedia Commons and is in the public domain. We use the 960-pixel thumbnail provided by Commons, without editing. Source page. Public domain mark. Blog figures The cover and the sketch of the multiplication $23\times14$ are original illustrations. The sketch explains the calculation and does not reproduce the mechanism of the historical machine. The interactive figures for the eight bits, the terms of the product rule and the sums of the series for π were created for this trail. The examples in the text let you follow the calculations even without JavaScript.