1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
3.
Numerical digit
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A digit is a numeric symbol used in combinations to represent numbers in positional numeral systems. The name digit comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 numeral system, i. e. the decimal digits. In a given system, if the base is an integer. For example, the system has ten digits, whereas binary has two digits. In a basic system, a numeral is a sequence of digits. Each position in the sequence has a value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, each digit in a number system represents an integer. For example, in decimal the digit 1 represents the one, and in the hexadecimal system. A positional number system must have a digit representing the integers from zero up to, but not including, thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals 0 to 9 in the rightmost units position. The Hindu–Arabic numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the place or units place. Each successive place to the left of this has a value equal to the place value of the previous digit times the base. Similarly, each place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10, the total value of the number is 1 ten,0 ones,3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, the place value of any given digit in a numeral can be given by a simple calculation, which in itself is a compliment to the logic behind numeral systems. And to the right, the digit is multiplied by the base raised by a negative n, for example, in the number 10. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet widely accepted. Instead of a zero, a dot was left in the numeral as a placeholder, the first widely acknowledged use of zero was in 876. The original numerals were very similar to the ones, even down to the glyphs used to represent digits
4.
Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base
5.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
6.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and 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 these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, 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 personally managed his sequences for almost 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, 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, besides integer sequences, the OEIS also 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, 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, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often 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, a is 1480028129, but there is no such 2×2 magic square, so a is 0
7.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
8.
Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5