Innovations of the Seventeenth Century 4 – Binary Numbers

Published on on June 6, 2010

While attributed to many including Gottfried Wilhelm Leibniz (1646-1716) and the Chinese, John Napier was the first to propose a binary number system and the accompanying arithmetic. Napier used a system of the letters of the alphabet to represent ascending powers of two.

Whilst mucking around with powers of various bases, it was inevitable that he would at some point stumble across binary mathematics. In an appendix to Rabdologiae, seu Numerationis per Virgulas Libri Duo titled Arithmeticae Localis, Napier shows how to convert between decimal and binary numbers and how to conduct binary mathematics.

Napier did not see any serious applications for his binary numbers.

“I came upon this Arithmetical Table, which must justly be called a game rather than hard work.”


Addition is carried out by combining the two numbers in his alphabetic binary notation in alphabetical order. Wherever two of the same letter appears, the two letters are replaced by one instance of the next letter in the alphabet. For example:

abd + d = abdd = abe

in modern notation, we’d write

1011 + 1000 = 10011


abd + e = abde (1011 + 10000 = 11011)


abd – d = ab (1011 – 1000 = 11)

Napier also showed how to use a grid marked with the both letters and powers of two up the side and across the bottom to carry out his operations. He used counters in positions where we would use the 1 in modern binary notation. This was particularly useful for longer operations such as multiplication and division.

Leibniz’s main contribution was to introduce the modern binary notation consisting of ones and zeros in 1679. Leibniz also didn’t see any particular application at the time.

More Reading

Electropaedia – History of Batteries (and other things)

Hawkins, W.F. Napier’s mathematical works – translations by W.F. Hawkins, University of Auckland, 1978

Napier, Rabdologiae, seu Numerationis per Virgulas Libri duo (1617)

The Greatest Mathematicians of All Time

This post was inspired by Gareth Cronin’s Uni Assignment on the Works of John Napier. The assignment can be seen here.


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