Editing Binary number (section)

===Leibniz===
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|upright|Gottfried Leibniz]]
Leibniz wrote in excess of a hundred manuscripts on binary, most of them remaining unpublished.<ref name=":0">{{Citation |last=Strickland |first=Lloyd |title=Leibniz on Number Systems |date=2020 |work=Handbook of the History and Philosophy of Mathematical Practice |pages=1–31 |editor-last=Sriraman |editor-first=Bharath |url=https://link.springer.com/referenceworkentry/10.1007/978-3-030-19071-2_90-1 |access-date=2024-08-20 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-19071-2_90-1 |isbn=978-3-030-19071-2}}</ref> Before his first dedicated work in 1679, numerous manuscripts feature early attempts to explore binary concepts, including tables of numbers and basic calculations, often scribbled in the margins of works unrelated to mathematics.<ref name=":0" />

His first known work on binary, ''“On the Binary Progression"'', in 1679, Leibniz introduced conversion between decimal and binary, along with algorithms for performing basic arithmetic operations such as addition, subtraction, multiplication, and division using binary numbers. He also developed a form of binary algebra to calculate the square of a six-digit number and to extract square roots.<ref name=":0" />

His most well known work appears in his article ''Explication de l'Arithmétique Binaire'' (published in 1703).
The full title of Leibniz's article is translated into English as the ''"Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of [[Fu Xi]]"''.<ref name="lnz">Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.[https://www.leibniz-translations.com/binary]</ref> Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows:<ref name="lnz" />
: 0 0 0 1 &nbsp; numerical value 2<sup>0</sup>
: 0 0 1 0 &nbsp; numerical value 2<sup>1</sup>
: 0 1 0 0 &nbsp; numerical value 2<sup>2</sup>
: 1 0 0 0 &nbsp; numerical value 2<sup>3</sup>

While corresponding with the Jesuit priest [[Joachim Bouvet]] in 1700, who had made himself an expert on the ''I Ching'' while a missionary in China, Leibniz explained his binary notation, and Bouvet demonstrated in his 1701 letters that the ''I Ching'' was an independent, parallel invention of binary notation.
Leibniz & Bouvet concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical [[mathematics]] he admired.<ref>[https://gwern.net/doc/cs/1980-swiderski.pdf#page=8 "Bouvet and Leibniz: A Scholarly Correspondence"], Swiderski 1980</ref> Of this parallel invention, Leibniz wrote in his "Explanation Of Binary Arithmetic" that "this restitution of their meaning, after such a great interval of time, will seem all the more curious."<ref>[https://www.leibniz-translations.com/binary.htm Leibniz]: "The Chinese lost the meaning of the Cova or Lineations of Fuxi, perhaps more than a thousand years ago, and they have written commentaries on the subject in which they have sought I know not what far out meanings, so that their true explanation now has to come from Europeans. Here is how: It was scarcely more than two years ago that I sent to Reverend Father Bouvet,<sup>3</sup> the celebrated French Jesuit who lives in Peking, my method of counting by 0 and 1, and nothing more was required to make him recognize that this was the key to the figures of Fuxi. Writing to me on 14 November 1701, he sent me this philosophical prince's grand figure, which goes up to 64, and leaves no further room to doubt the truth of our interpretation, such that it can be said that this Father has deciphered the enigma of Fuxi, with the help of what I had communicated to him. And as these figures are perhaps the most ancient monument of [GM VII, p227] science which exists in the world, this restitution of their meaning, after such a great interval of time, will seem all the more curious."</ref>

The relation was a central idea to his universal concept of a language or [[characteristica universalis]], a popular idea that would be followed closely by his successors such as [[Gottlob Frege]] and [[George Boole]] in forming [[Propositional Calculus|modern symbolic logic]].<ref>{{Cite book
|last=Aiton
|first=Eric J.
|title=Leibniz: A Biography
|year=1985
|publisher=Taylor & Francis
|isbn=0-85274-470-6
|pages=245–8
}}</ref>
Leibniz was first introduced to the ''[[I Ching]]'' through his contact with the French Jesuit [[Joachim Bouvet]], who visited China in 1685 as a missionary. Leibniz saw the ''I Ching'' hexagrams as an affirmation of the [[Universality (philosophy)|universality]] of his own religious beliefs as a Christian.<ref name="smith">{{cite book|author1=J.E.H. Smith|title=Leibniz: What Kind of Rationalist?: What Kind of Rationalist?|url=https://books.google.com/books?id=Da_oP3sJs1oC&pg=PA4153|year=2008|publisher=Springer|isbn=978-1-4020-8668-7|page=415}}</ref> Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of ''[[creatio ex nihilo]]'' or creation out of nothing.<ref name="lniz">{{cite book|author1=Yuen-Ting Lai|title=Leibniz, Mysticism and Religion|url=https://books.google.com/books?id=U9dOmVt81UAC&pg=PA149|year=1998|publisher=Springer|isbn=978-0-7923-5223-5|pages=149–150}}</ref>

{{quote|[A concept that] is not easy to impart to the pagans, is the creation ''ex nihilo'' through God's almighty power. Now one can say that nothing in the world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing.|Leibniz's letter to the [[Rudolph Augustus, Duke of Brunswick-Lüneburg|Duke of Brunswick]] attached with the ''I Ching'' hexagrams<ref name="smith"/>}}