by Paul Curzon, Queen Mary University of London
I Ching the ancient Chinese divination text, several thousand years old, is based on a binary pattern…
I Ching is one of the oldest Chinese texts. The earliest copies date from around 3000 years ago. It uses 64 hexagrams: symbols consisting of six rows of lines (see right). Each row is either a solid horizontal line or two shorter lines with a gap in the middle. The 64 hexagrams are all the possible symbols that can be made from six rows of lines in this way. In I Ching, they each represent possible predictions about the future (a bit like horoscopes). To use I Ching, a series of hexagrams were chosen. This was done in some unknown but random way, using stalks of the Yarrow plant, standing in for dice. The chosen hexagrams then told the person something about their future.
In the earliest versions of I Ching, the order of the hexagrams suggests that they were not thought of as numbers as such. However, in a later version, from around 1000 AD the order in which they appear is different. Thought to be written by a Chinese scholar, Shao Y\0x014Dng, it is this version that Leibniz was given and that aroused his interest because the order of the hexagrams follows the pattern we know of as binary (see Predicting the future). Shao Y\0x014Dng had apparently picked the sequence deliberately because of the binary pattern, so understood it as a counting sequence, if not necessarily how to do maths with it.
How do the hexagrams correspond to binary? It is not in the lines themselves but the pattern of line breaks down the middle that matters. Think of a break in the lines as a 0 (yin) and no break as a 1 (yang). The order, as Leibniz realised, is a counting system, equivalent to our decimals but where you only have two digits (0 and 1) rather than our ten digits (0…9).
Whereas in decimal each column of a number like 123 represents a power of 10 (ones, tens, hundreds, …) in binary each column represents a power of 2 (ones, twos, fours, eights, …). To work out the value that the number represents you multiply each digit by its column value and add the results. So in decimal, 123 represents one hundred plus two tens plus three ones (one hundred and twenty three). 1011 in decimal represents one thousand plus no hundreds plus one ten plus a one (one thousand and eleven). In binary, however, 1011 represents instead one eight plus no fours plus one two plus a one (8+0+2+1) so eleven. It is just a different way of writing down numbers.
Investigating the I Ching pattern helped Leibniz to work out the mathematics of binary arithmetic and on to thinking about machines to do calculations using it.
More on …
- Predicting the future: marbles, binary and the I Ching
- Cunning Computational Contraptions
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This article was funded by UKRI, through Professor Ursula Martin’s grant EP/K040251/2 and grant EP/W033615/1.