Neil Sloane

Neil James Alexander Sloane is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, sphere packing. Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences. Sloane was brought up in Australia, he studied at Cornell University under Nick DeClaris, Frank Rosenblatt, Frederick Jelinek and Wolfgang Heinrich Johannes Fuchs, receiving his Ph. D. in 1967. His doctoral dissertation was titled Lengths of Cycle Times in Random Neural Networks. Sloane joined AT&T Bell Labs in 1968 and retired from AT&T Labs in 2012, he became an AT&T Fellow in 1998. He is a Fellow of the Learned Society of Wales, an IEEE Fellow, a Fellow of the American Mathematical Society, a member of the National Academy of Engineering, he is a winner of a Lester R. Ford Award in 1978 and the Chauvenet Prize in 1979. In 2005 Sloane received the IEEE Richard W. Hamming Medal. In 2008 he received the Mathematical Association of America David P. Robbins award, in 2013 the George Pólya Award.

In 2014, to celebrate his 75th birthday, Sloane shared some of his favorite integer sequences. Besides mathematics, he has authored two rock-climbing guides to New Jersey. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY, 1973. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North-Holland, Amsterdam, 1977. M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, San Diego CA, 1979. N. J. A. Sloane and A. D. Wyner, Claude Elwood Shannon: Collected Papers, IEEE Press, NY, 1993. N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995. J. H. Conway and N. J. A. Sloane, Sphere Packings and Groups, Springer-Verlag, NY, 1st edn. 1988. A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, Springer-Verlag, NY, 1999. G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer-Verlag, 2006. Reeds–Sloane algorithm Neil Sloane at the Mathematics Genealogy Project IEEE Richard W. Hamming Medal Recipients, 2005 – Neil J. A. Sloane Neil Sloane's entry in the Numericana Hall of Fame "The pattern collector", Science News Doron Zeilberger, Opinion 124: A Database is Worth a Thousand Mathematical Articles

Prime number

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.

Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.

Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.

Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.

The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide

Hexadecimal

In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most the symbols "0"–"9" to represent values zero to nine, "A"–"F" to represent values ten to fifteen. Hexadecimal numerals are used by computer system designers and programmers, as they provide a more human-friendly representation of binary-coded values; each hexadecimal digit represents four binary digits known as a nibble, half a byte. For example, a single byte can have values ranging from 0000 0000 to 1111 1111 in binary form, which can be more conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is used to specify the radix. For example the decimal value 10,995 would be expressed in hexadecimal as 2AF316. In programming, a number of notations are used to support hexadecimal representation involving a prefix or suffix; the prefix 0x is used in C and related languages, which would denote this value by 0x2AF3. Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.

In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript can give the base explicitly: 15910 is decimal 159; some authors prefer a text subscript, such as 159decimal and 159hex, or 159h. In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen: In URIs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space character, ASCII code point 20 in hex, 32 in decimal. In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation ode, thus ’. In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #: white, for example, is represented #FFFFFF.

CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33. Unix shells, AT&T assembly language and the C programming language use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits:'\x1B' represents the Esc control character. To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In MIME quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits prefixed by an equal to sign =, as in Espa=F1a to send "España". In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h: FFh or 05A3H; some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh Other assembly languages, Delphi, some versions of BASIC, GameMaker Language and Forth use $ as a prefix: $5A3.

Some assembly languages use the notation H'ABCD'. Fortran 95 uses Z'ABCD'. Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3". Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant; the Smalltalk language uses the prefix 16r: 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC... Common Lisp uses the prefixes # 16r. Setting the variables *read-base* and *print-base* to 16 can be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers, thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16. MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3 BBC BASIC and Locomotive BASIC use & for hex.

TI-89 and 92 series uses a 0h prefix: 0h5A3 ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary and octal numbers can be specified similarly; the most common format for hexadecimal on IBM mainframes and midrange computers running the traditional OS's is X'5A3', is used in Assembler, PL/I, COBOL, JCL, scripts and other places. This format was common on

Roman numerals

The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, employ seven symbols, each with a fixed integer value, as follows: The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by the more convenient Arabic numerals; the original pattern for Roman numerals used the symbols I, V, X as simple tally marks. Each marker for 1 added a unit value up to 5, was added to to make the numbers from 6 to 9: I, II, III, IIII, V, VI, VII, VIII, VIIII, X; the numerals for 4 and 9 proved problematic, are replaced with IV and IX. This feature of Roman numerals is called subtractive notation; the numbers from 1 to 10 are expressed in Roman numerals as follows: I, II, III, IV, V, VI, VII, VIII, IX, X.

The system being decimal and hundreds follow the same underlying pattern. This is the key to understanding Roman numerals: Thus 10 to 100: X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C. Note that 40 and 90 follow the same subtractive pattern as 4 and 9, avoiding the confusing XXXX. 100 to 1000: C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M. Again - 400 and 900 follow the standard subtractive pattern, avoiding CCCC. In the absence of standard symbols for 5,000 and 10,000 the pattern breaks down at this point - in modern usage M is repeated up to three times; the Romans had several ways to indicate larger numbers, but for practical purposes Roman Numerals for numbers larger than 3,999 are if used nowadays, this suffices. M, MM, MMM. Many numbers include hundreds and tens; the Roman numeral system being decimal, each power of ten is added in descending sequence from left to right, as with Arabic numerals. For example: 39 = "Thirty nine" = XXXIX. 246 = "Two hundred and forty six" = CCXLVI. 421 = "Four hundred and twenty one" = CDXXI.

As each power of ten has its own notation there is no need for place keeping zeros, so "missing places" are ignored, as in Latin speech, thus: 160 = "One hundred and sixty" = CLX 207 = "Two hundred and seven" = CCVII 1066 = "A thousand and sixty six" = MLXVI. Roman numerals for large numbers are nowadays seen in the form of year numbers, as in these examples: 1776 = MDCCLXXVI. 1954 = MCMLIV 1990 = MCMXC. 2014 = MMXIV (the year of the games of the XXII Olympic Winter Games The current year is MMXIX. The "standard" forms described above reflect typical modern usage rather than an unchanging and universally accepted convention. Usage in ancient Rome varied and remained inconsistent in medieval times. There is still no official "binding" standard, which makes the elaborate "rules" used in some sources to distinguish between "correct" and "incorrect" forms problematic. "Classical" inscriptions not infrequently use IIII for "4" instead of IV. Other "non-subtractive" forms, such as VIIII for IX, are sometimes seen, although they are less common.

On the numbered gates to the colosseum, for instance, IV is systematically avoided in favour of IIII, but other "subtractives" apply, so that gate 44 is labelled XLIIII. Isaac Asimov speculates that the use of "IV", as the initial letters of "IVPITER" may have been felt to have been impious in this context. Clock faces that use Roman numerals show IIII for four o'clock but IX for nine o'clock, a practice that goes back to early clocks such as the Wells Cathedral clock of the late 14th century. However, this is far from universal: for example, the clock on the Palace of Westminster, Big Ben, uses a "normal" IV. XIIX or IIXX are sometimes used for "18" instead of XVIII; the Latin word for "eighteen" is rendered as the equivalent of "two less than twenty" which may be the source of this usage. The standard forms for 98 and 99 are XCVIII and XCIX, as described in the "decimal pattern" section above, but these numbers are rendered as IIC and IC originally from the Latin duodecentum and undecentum.

Sometimes V and L are not used, with instances such as IIIIII and XXXXXX rather than VI or LX. Most non-standard numerals other than those described above - such as VXL for 45, instead of the standard XLV are modern and may be due to error rather than being genuine variant usage. In the early years of the 20th century, different representations of 900 appeared in several inscribed dates. For instance, 1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, while on the north entrance to the Saint Louis Art Museum, 1903 is inscribed as MDCDIII rather than MCMIII. Although Roman numerals came to be written with letters

Negative number

In mathematics, a negative number is a real number, less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level negative represents below sea level. If positive represents a deposit, negative represents a withdrawal, they are used to represent the magnitude of a loss or deficiency. A debt, owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature; the laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.

For example, − = 3 because the opposite of an opposite is the original value. Negative numbers are written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number the negative sign is placed higher than the minus sign. Conversely, a number, greater than zero is called positive; the positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign; every real number other than zero is either negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers are referred to as integers. In bookkeeping, amounts owed are represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty, but may well contain much older material.

Liu Hui established rules for subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of negative numbers around the middle of the 19th century. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd; some mathematicians like Leibniz agreed that negative numbers were false, but still used them in calculations. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: 0 − 3 = −3. In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers.

For example, 5 − 8 = −3since 8 − 5 = 3. The relationship between negative numbers, positive numbers, zero is expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less, thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example though 8 is greater than 5, written 8 > 5negative 8 is considered to be less than negative 5: −8 < −5. It follows that any negative number is less than any positive number, so −8 < 5 and −5 < 8. In the context of negative numbers, a number, greater than zero is referred to as positive, thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number, either positive or zero, while nonpositive is used to refer to a number, either negative or zero.

Zero is a neutral number. Goal difference in association football and hockey. Plus-minus differential in ice hockey: the difference in total goals scored for the team and against the team when a particular player is on the ice is the player’s +/− rating. Players can have a negative rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. British football clubs are deducted points if they enter administration, thus have a negative points total until they have earned at least that many points that season. Lap times in Formula 1 may be given as the difference compared to a previous lap, will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorde

On-Line Encyclopedia of Integer Sequences

The On-Line Encyclopedia of Integer Sequences cited as Sloane's, is an online database of integer sequences. It was maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, is cited; as of September 2018 it contains over 300,000 sequences. Each entry contains the leading terms of the sequence, mathematical motivations, literature links, more, including the option to generate a graph or play a musical representation of the sequence; the database is searchable by subsequence. Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics; the database was at first stored on punched cards.

He published selections from the database in book form twice: A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487; the Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not. These books were well received and after the second publication, mathematicians supplied Sloane with a steady flow of new sequences; the collection became unmanageable in book form, when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, soon after as a web site. As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998; the database continues to grow at a rate of some 10,000 entries a year.

Sloane has managed'his' sequences for 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, was added to the database in November 2011. Besides integer sequences, the OEIS catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences: the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.

Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, it still uses a linear form of conventional mathematical notation. Greek letters are represented by their full names, e.g. mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits always referred to with leading zeros, e.g. A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by periods, or spaces. In comments, etc. A represents the nth term of the sequence. Zero is used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2. But there is no such 2×2 magic square, so a is 0; this special usage has a solid mathematical basis in certain counting functions.

For example, the totient valence function. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions. −1 is used for this purpose instead, as in A094076. The OEIS ma

Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: "0" and "1". The base-2 numeral system is a positional notation with a radix of 2; each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by all modern computers and computer-based devices; the modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt and India. Leibniz was inspired by the Chinese I Ching; the scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.

Early forms of this system can be found in documents from the Fifth Dynasty of Egypt 2400 BC, its developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt 1200 BC. The method used for ancient Egyptian multiplication is closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value is either doubled or has the first number added back into it; this method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China; the binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang.eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.

Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Indian scholar Pingala developed a binary system for describing prosody, he used binary numbers in the form of long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter; the binary representations in Pingala's system increases towards the right, not to the left like in the binary numbers of the modern, Western positional notation. The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Asia. Sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.

In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or ‘Ars generalis’ based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could be encoded as scarcely visible variations in the font in any random text. For the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only. John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results.

The first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679. Leibniz's system uses 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 0 0 0 1 numerical value 20 0 0 1 0 numerical value 21 0 1 0 0 numerical value 22 1 0 0 0 numerical value 23Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus; as a Sinophile, Leibniz was aware of