27 Mar 2009

Leibniz' Representation and Computation of Binary Numerals

by Corry Shores
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[The following is based on Lloyd Strickland's online translation of Leibniz' "Explanation of Binary Arithmetic"]

Leibniz' Representation and Computation of Binary Numerals

We normally reckon with our ten decimal digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
When we reach ten, we start the series over at 1 in the next decimal place, and at zero in the first decimal place, so ten is rendered '10.'

Leibniz finds that a positional system with a base two is the "simplest progression of all," that is "useful for the perfection of the science of numbers." We use just 0 and 1. When reaching two, we start again, just as we normally do when we reach ten in the decimal system.

So zero is 0
We add one to get
We add one again, which gives us two. But we do not have a single symbol for two. Rather, we carry to the next place, just as if we were adding one to nine in the decimal system. So

So the binary digit '10' is the same as the decimal digit '2.' Now let's add another one to our binary two, which is rendered '10.'

We see that in binary, the rendition of three is '11.' We will add another one to get four.

So four is rendered '100.' Hence we see Leibniz' table for binary numbers. The right column are the numbers as we know them in decimal digits. The broad left column is their binary rendition.

Now, we know that four is 100, and two is 10. So let's add two and two together. We will see that we do the same operations as with decimal digits, only we carry at two and not at ten.

Now consider Leibniz' examples of mathematical operations using binary.

Images from the original text:

Images and text references from the English translation:
Leibniz, "Explanation of Binary Arithmetic, which Uses only the Characters 0 and 1, with some Remarks on Its Usefulness, and on the Light It Throws on the Ancient Chinese Figures of Fuxi." Tranls. Lloyd Strickland. 2007.
Original text, "Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 et 1, avec des remarques sur son utilité, et sur ce qu'elle donne le sens des anciennes figures chinoises de fohy." In Die mathematische schriften von Gottfried Wilheim Leibniz, vol. VII. C. I. Gerhardt (ed), Halle: Druck und Verlag von H. W. Schmidt, 1863, pp 223-227.
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