### Binary system

The binary (numeral) system (or base 2 numerals)  is a positional numeral system with a radix of 2. It represents numeric values using two symbols and .

The modern binary number system goes back to Gottfried Leibniz who in the 17th century proposed and developed it in his article Explication de l'Arithmétique Binaire   [1]   . Leibniz invented the system around 1679 but he published it in 1703. He already used symbols 0 and 1. About the binary calculations he wrote ...these operations are so easy that we shall never have to guess ar apply trial and error, as we must do in ordinary division. Nor do we need to learn anything by rote... Leibniz even proposed the Duke of Brunswick to issue a silver medal commemorating this discovery with the following inscriptions: The model of creation discovered by G.W.L.  and   One is enough for deriving everything from nothing. Simon Marquis de Laplace wrote: Laibnitz saw in his binary arithmetic the image of Creation ... . He imaginated that Unity represents God, and Zero the Void, that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration. ...

Actually the first application of the binary system is essentially older. The ancient Egyptians used for multiplication of two numbers a procedure today known as the peasant multiplication which basis is the expression of one factor in the binary system. Since they were not interested in finding proofs justifying the used procedures, they were not awaken to this connection.

In Europe it was Thomas Harriot (or Hariot or Harriott) (c. 1560 - July 2, 1621), an English astronomer, mathematician, ethnographer, linguist and the founder of the English school of algebra, who discovered the binary system. In the eight large volumes of Hariot’s manuscripts kept in the British Museum fragmentary calculations, with occasional connected notes on a diversity of subjects can be found. One page, otherwise blank contains [2] (see also [3] , [4] ):

In 1605 Francis Bacon (22 January 1561 - 9 April 1626) English philosopher and statesman, developed a biliteral steganography method (a method of hiding a secret message as opposed to a true cipher) in which each letter of the plain text is replaced by a group of five of the letters ‘a’ or ‘b’. Actually he prepared a cipher scheme for handwritten capital and small letters with each having two alternative forms, one to be used as and the other as (see an illustrated plate in his De Augmentis Scientiarum (The Advancement of Learning), pp. 266-270).  Every letter in the extended alphabet was represented by a specific configuration of these two letters in such a way that by replacing with zero and with 1, the letters corresponded to the numbers 0 to 26 in our present-day binary notation. This is a predecessor of the ASCI code. Bacon’s code is given by the table:

To our knowledge the first published treatment of the binary system appeared  ([5] , p. 199)  in the work of a Cistercian bishop Juan  Caramuel y Lobkowitz   . His  Mathesis biceps: Vetus et nova (Two-headed mathematics: old and new) [6]  was considered as the largest mathematical encyclopedia of his time. On pp. 45-48 he discussed the representation of numbers in radices 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 and 60 at some length but giving examples of arithmetic operations only in decimal or hexadecimal systems.

The advantage of the binary system is that it can be used to represent numbers in systems (mechanisms) which are capable of being in two mutually exclusive states. This is the reason that the binary system is used internally by all modern computers. However the first prototype binary computer was built by John Atanasoff, a physics professor at Iowa State College, in 1939. The Frenchman Raymond Louis André Valtat from Paris patented in 1932 in Germany a calculator predicted on the binary system. In his paper  [7] he advocated the usage of  the binary system in a calculating apparatus in comparison to the decimal system, for instance that the computation of a square root is especially simple in this system.  In 1936 Alan Turing designed an electromechanical multiplier. In 1938, the American George Stibitz built an binary adder using electromechanical relays. In 1946 A.W.Burks, H.H.Goldstine 1  and J. von Neumann published a memorandum  [8] in which they advocate abandoning the decimal representation in favor of the binary system.

One way to transfer decimal numbers to binary system is by a process of successive division (the so-called division by 2 using radix 10 arithmetic). For instance, to transform the integer 125 from decimal to binary, the number is iteratively divided by 2 and the remainders of the successive divisions are stored. The iterative process ends when the quotient 0 is reached.

Thus the representation of the decimal number 125 in binary numeral system is  1111101.

The following modification of the previous algorithm can be found in Legenedre’s  [9] , p. 229, or  [10] , footnote on p.  244. The division is by 64 and the remainders are converted from radix 10 representation to binary ones.

Therefore .  To convert more positive integers from decimal to binary visit .

To convert from binary to decimal is the reverse algorithm. For instance, to convert to decimal form we can use the template

and simply add the powers of 2 standing over ones, that is .  A more efficient way of calculation is to proceed as follows . This is actually a form of Horner’s rule which in this case is more known informally as the snake method.

The idea behind the graphical representation is to always alternate horizontal and vertical moves, and adding on vertical moves and doubling on horizontal moves. To convert more positive integers from binary to decimal form visit .

The mentioned division by 2 using radix 10 arithmetic method can be also interpreted in form of the snake method as an algorithm to convert a number from its decimal representation to its binary representation. We now proceed from the left to the right. Going up or down corresponds to subtraction of the least non-negative remainder modulo 2 and going left corresponds to division by 2.

### Notes

 1 Herman Heine Goldstine (September 13, 1913 - June 16, 2004),  American computer scientist, one of the original developers of first modern electronic digital computer ENIAC.

### References

 [1] Leibniz, G. W. (1703). 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 (An explanation of binary arithmetic using the characters 0 and 1, with remarks about its utility and the meaning it gives to the ancient Chinese figures of Fuxi). Memoires de l’Académie Royale des Sciences, 3, 85-93.

 [2] Morley, F. V. (1922). Thomas Hariot--1560-1621. The Scientific Monthly, 14(1), 60-66.

 [3] Glaser, A. (1981). History of binary and other nondecimal numeration (Rev. ed.). Los Angeles: Tomash Publishers.

 [4] Shirley, J. W. (1951). Binary numeration before Leibniz. Am. J. Phys., 19, 452-454.

 [5] Knuth, D. E. (1998). The Art of Computer Programming. Vol. 2, Seminumerical Algorithms. Boston San Francisco New York etc.: Addison-Wesley.

 [6] y Lobkowitz , J. C. (1670). Mathesis biceps, vetus, et nova in omnibus, et singulis Veterum, et Recentiorum Placita examinantur ; interdum corriguntur, semper dilucidantur : et pleraque omnia Mathemata reducuntur speculative et practice ad facillimos, et expeditissimos Canones. Accedent alii tomi videlicet: Architectvra recta ... Architectvra obliqva ... Architectvra militaris ... Mvsica ... Astronomia physica, 2 Vols.. Campaniae: In Officina Episcopali. Prostant Lugduni apud Laurentium Anisson..

 [7] Valtat, R. L. (1936). Machine à calculer fondée sur l'emploi de la numération binaire. (French). C. R. Acad. Sci., Paris, , 202, 1745-1747.

 [8] Burks, A. W., Goldstine, H. H., & von Neumann, J. (1946). Preliminary Discussion of the Logical Design of an Electronic Computer Instrument. Institute for Advanced Study. ASIN B0007HW8WE,

 [9] Legendre, A. M. (1798). Essai sur la théorie des nombres. Paris: Duprat.

 [10] Legendre, A. M. (1886). Zahlentheorie (nach 3. Aufl. ins deutsche übertragen von H. Maser). Leipzig: B.G.Teubner.

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Štefan Porubský: Binary system.

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