MadSci Network: Science History

Re: What is the origin of binary numbers?

Date: Sun Feb 10 12:58:47 2002
Posted By: Mark Huber, Post-doc/Fellow, Statistics, Stanford University
Area of science: Science History
ID: 1012606134.Sh

Question: What is the origin of binary numbers?

Is it true That Leibniz came up with binary numbers upon studying the I ching, which supposedly originated from a fish man in china named FUXI?

There is of course nothing special about the decimal number system with its ten digits. The ancient Babylonians and Sumerians used a base 60 system as early as the 19th century B.C.E. Europeans learned decimal numbers and arithmetic from the Arab world, which in turn learned it from India. Various theories have India inventing the numbers, or learning it from the Chinese or Babylonians via the Greeks. In any case, the decimal numbers were widely used in Europe by the time of the Renaissance, and by that time no one questioned the use of a base 10 system.

No one, that is, until Leibniz came along. In 1679, he first realized that two digits—0 and 1—are all that are really needed for a positional number system. Binary arithmetic could handle any problem decimal arithmetic could deal with. At this point, Leibniz hadn't seen the I Ching.

The I Ching is a system where coins are used to randomly generate hexagrams, a group of six lines where each line represents either "yin" or "yang". Therefore, each hexagram can be thought of as a six digit number in the binary system. Since 2^6=64, there are 64 such hexagrams.

This site indicates that roughly ten years later, Leibniz was introduced to the I Ching by correspondence with Father Joachim Bouvet. Leibniz definitely had at least one book by Bouvet in his library, and Bouvet—a Jesuit missionary in China—apparently had access to the court of the Chinese Emperor. Certainly if Leibniz was introduced to the I Ching at this point, he would have immediately recognized it's relationship to binary numbers. Several authors (such as Umberto Eco here) and websites all point to Bouvet writing to Leibniz describing the I Ching, however, none of them presented a quote of Leibniz indicating that he ever knew of the I Ching, and none provided a historical reference. In any event, Leibniz knew about binary numbers before he heard of the I Ching.

Now the origin of the I Ching is murky at best, and is definitely the domain of mythology rather than historical fact. Some sites have the creator being FuXi the first Emperor of China, whereas this one represents FuXi as the first male being—with a human head and the body of a dragon. More prosaic is this view of FuXi as a set of advisors to a matriarchal tribe in ancient northwestern China.

What's fascinating about the I Ching (and with other systems such as Tarot) is that a few simple objects (in the case of the I Ching six coins) can be used to generate a large number of possibilities. Mathematically, the number of outcomes grows exponentially with the number of objects involved, and that makes using the I Ching both quick and entertaining.

Mark Huber

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