Leibniz's binary arithmetics

In the seventeenth century, Francis Bacon used binary 5-tuples as a code, but binary arithmetic as we currently understand it—doing actual arithmetic with binary numbers rather than just using binary representations—starts with Leibniz about 1679, though he didn't publicize it until the late 1600s. He heard about the Fu-Hsi ordering of the I-Ching hexagrams from Jesuit missionaries in China in 1701 and wrote a good deal about it thereafter (see Figures 1 and 2 and p. 219).

Figure 1. The title page of Leibniz's booklet [Leibniz 1734] explaining binary notation to a nobleman shows a medallion he created, later borrowed by the Stadtsparkasse of Hanover to honor Leibniz himself.

Figure 2. Leibniz's first writing on binary arithmetic, dated 11 March 1679.

However, Leibniz was anticipated by Thomas Harriot, 1604, who did not publish, and by John Napier, whose Rabdologiæ of 1617 gave binary arithmetic as far as computing square roots, but this seems to have been ignored.

But binary ideas go much further back. Some simple counting systems are more or less base 2 and there are many instances of duality in nature—hands, sexes, etc. But we are interested in material that is somewhat more mathematical.

Binary multiplication

The earliest implicit use of binary representations occurs in ancient Egyptian mathematics. Figure 3 is Problem 30 of the Rhind Mathematical Papyrus, ca. 1700 B.C.E., computing (⅔ + ⅒) ×13. The problem is to solve (⅔ + ⅒)x =10, which is being done by false position, using x = 13 as a trial. Because of their complicated notation for numbers, especially fractions, they multiplied by repeatedly doubling, then adding the appropriate terms. For instance, to multiply a number by 13, they computed successively the double, the quadruple and the octuple of the number, then added the number to its quadruple and its octuple.