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Plaindrome

A plaindrome is a number whose hexadecimal digits are in nondecreasing order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, ... (OEIS A023757).A number that is not a plaindrome is called a katadrome.The following table summarizes related classes of numbers.namebase-16 digit orderkatadromestrict descendingmetadromestrict ascendingnialpdromenonincreasingplaindromenondecreasing

Decimal expansion

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number with decimal expansion 1234.56 is defined as(1)(2)Expressions written in this form (where negative are allowed as exemplified above but usually not considered in elementary education contexts) are said to be in expanded notation.Other examples include the decimal expansion of given by 625, of given by 3.14159..., and of given by 0.1111.... The decimal expansion of a number can be found in the Wolfram Language using the command RealDigits[n], or equivalently, RealDigits[n, 10].The decimal expansion of a number may terminate (in which case the number is called a regular number or finite decimal, e.g., ), eventually..

Octal

The base 8 notational system for representing real numbers. The digits used are 0, 1, 2, 3, 4, 5, 6, and 7, so that (8 in base 10) is represented as () in base 8. The following table gives the octal equivalents of the first few decimal numbers.11111321252212142226331315232744141624305515172531661620263277172127338101822283491119232935101220243036The song "New Math" by Tom Lehrer (That Was the Year That Was, 1965) explains how to compute in octal. (The answer is .)

Number length

The length of a number in base is the number of digits in the base- numeral for , given by the formulawhere is the floor function.The multiplicative persistence of an -digit is sometimes also called its length.

Binary

The base 2 method of counting in which only the digits 0 and 1 are used. In this base, the number 1011 equals . This base is used in computers, since all numbers can be simply represented as a string of electrically pulsed ons and offs. In computer parlance, one binary digit is called a bit, two digits are called a crumb, four digits are called a nibble, and eight digits are called a byte.An integer may be represented in binary in the Wolfram Language using the command BaseForm[n, 2], and the first digits of a real number may be obtained in binary using RealDigits[x, 2, d]. Finally, a list of binary digits can be converted to a decimal rational number or integer using FromDigits[l, 2].The illustration above shows the binary numbers from 0 to 63 represented graphically (Wolfram 2002, p. 117), and the following table gives the binary equivalents of the first few decimal numbers.1111101121101012101211002210110311131101231011141001411102411000510115111125110016110161000026110107111171000127110118100018100102811100910011910011291110110101020101003011110A..

Nialpdrome

A nialpdrome is a number whose hexadecimal digits are in nonincreasing order. The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 48, 49, 50, ... (OEIS A023771), corresponding to 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 20, 21, 22, 30, 31, 32, ....A number that is not a nialpdrome is a metadrome.The following table summarized related classes of numbers.namebase-16 digit orderkatadromestrict descendingmetadromestrict ascendingnialpdromenonincreasingplaindromenondecreasing

Midy's theorem

If the period of a repeating decimal for , where is prime and is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example, , and .

Cyclic number

A cyclic number is an -digit integer that, when multiplied by 1, 2, 3, ..., , produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913).The decimal expansions giving the first fewcyclic numbers are(1)(2)(3)(4)(OEIS A004042).The numbers of cyclic numbers for , 1, 2, ... are 0, 1, 9, 60, 467, 3617, 25883, 248881, 2165288, 19016617, 170169241, ... (OEIS A086018). It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist. In fact, the fraction of cyclic numbers out of all primes has been conjectured to be Artin's constant . The fraction of cyclic numbers among primes is 0.3739551.When a cyclic number is multiplied by its generator, the result is a string of 9s.This is a special case of Midy's theorem.See Yates (1973) for a table of prime period lengths for primes ...

Full reptend prime

A prime for which has a maximal period decimal expansion of digits. Full reptend primes are sometimes also called long primes (Conway and Guy 1996, pp. 157-163 and 166-171). There is a surprising connection between full reptend primes and Fermat primes.A prime is full reptend iff 10 is a primitive root modulo , which means that(1)for and no less than this. In other words, the multiplicative order of (mod 10) is . For example, 7 is a full reptend prime since .The full reptend primes are 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ... (OEIS A001913). The first few decimal expansions of these are(2)(3)(4)(5)Here, the numbers 142857, 5882352941176470, 526315789473684210, ... (OEIS A004042) corresponding to the periodic parts of these decimal expansions are called cyclic numbers. No general method is known for finding full reptend primes.The number of full reptend primes less than for , 2, ... are 1, 9, 60, 467, 3617, ... (OEIS A086018).A..

Least significant bit

The value of the bit in a binary number. For the sequence of numbers 1, 2, 3, 4, ..., the least significant bits are therefore the alternating sequence 1, 0, 1, 0, 1, 0, ... (OEIS A000035). It can be represented as(1)(2)or(3)It is also given by the linear recurrenceequation(4)with (Wolfram 2002, p. 128).Analogously, the "most significant bit" is the value of the bit in an -bit representation.The least significant bit has Lambert series(5)where is a q-polygamma function.

Unique prime

Following Yates (1980), a prime such that is a repeating decimal with decimal period shared with no other prime is called a unique prime. For example, 3, 11, 37, and 101 are unique primes, since they are the only primes with periods one (), two (), three (), and four () respectively. On the other hand, 41 and 271 both have period five, so neither is a unique prime.The unique primes are the primes such thatwhere is a cyclotomic polynomial, is the period of the unique prime, is the greatest common divisor, and is a positive integer.The first few unique primes are 3, 11, 37, 101, 9091, 9901, 333667, ... (OEIS A040017), which have periods 1, 2, 3, 4, 10, 12, 9, 14, 24, ... (OEIS A051627), respectively.

Hexadecimal

The base 16 notational system for representing real numbers. The digits used to represent numbers using hexadecimal notation are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The following table gives the hexadecimal equivalents for decimal numbers from 1 to 30.1111B21152212C22163313D23174414E24185515F2519661610261A771711271B881812281C991913291D10A2014301EThe hexadecimal system is particularly important in computer programming, since four bits (each consisting of a one or zero) can be succinctly expressed using a single hexadecimal digit. Two hexadecimal digits represent numbers from 0 to 255, a common range used, for example, to specify colors. Thus, in the HTML language of the web, colors are specified using three pairs of hexadecimal digits RRGGBB, where is the amount of red, the amount of green, and the amount of blue.In hexadecimal, numbers with increasing digits are called metadromes, those with nondecreasing digits..

Hereditary representation

The representation of a number as a sum of powers of a base , followed by expression of each of the exponents as a sum of powers of , etc., until the process stops. For example, the hereditary representation of 266 in base 2 is(1)(2)

Repeating decimal

A repeating decimal, also called a recurring decimal, is a number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely). The repeating portion of a decimal expansion is conventionally denoted with a vinculum so, for example,The minimum number of digits that repeats in such a number is known as the decimalperiod.Repeating decimal notation was implemented in versions of the Wolfram Language prior to 6 as PeriodicForm[RealDigits[r]] after loading the add-on package NumberTheory`ContinuedFractions`.All rational numbers have either finite decimal expansions (i.e., are regular numbers; e.g., ) or repeating decimals (e.g., ). However, irrational numbers, such as neither terminate nor become periodic.Numbers such as 0.5 are sometimes regarded as repeating decimals since .The denominators of the first few unit fractions having repeating decimals are 3, 6, 7, 9, 11, 12, 13,..

Regular number

A regular number, also called a finite decimal (Havil 2003, p. 25), is a positive number that has a finite decimal expansion. A number such as which is not regular is said to be nonregular.If is a regular number, then(1)(2)(3)Factoring possible common multiples gives(4)where (mod 2, 5).The denominators of the first few regular unit fractions are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, ... (OEIS A003592), which have decimal expansions 1, , , , , , , , , , , , ....The number of decimal digits in a regular number is given by (Wells 1986, p. 60). The numbers of digits in the regular unit fractions are 1, 2, 1, 3, 1, 4, 2, 2, 5, 3, 2, 6, 4, 2, 3, ... (OEIS A117920).

Figures

A number is said to have " figures" if it takes digits to express it. The number of figures is therefore equal to one more than the power of 10 in the scientific notation representation of the number. The word is most frequently used in reference to monetary amounts, e.g., a "six-figure salary" would fall in the range of to .

Quaternary

The base-4 method of counting in which only the digits 0, 1, 2, and 3 are used. The illustration above shows the numbers 0 to 63 represented in quaternary, and the following table gives the quaternary equivalents of the first few decimal numbers.111123211112212302211233133123113410143224120511153325121612161002612271317101271238201810228130921191032913110222011030132These digits have the following multiplicationtable.0123000001012320210123031221

Euler's totient rule

The number of bases in which is a repeating decimal (actually, repeating -ary) of length is the same as the number of fractions , , ..., which have reduced denominator . For example, in bases 2, 3, ..., 6, 1/7 is given by(1)(2)(3)(4)(5)which have periods 3, 6, 3, 6, and 2, respectively, corresponding to the denominators6, 3, 2, 3, and 6 of(6)

Duodecimal

The base-12 number system composed of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B. Such a system has been advocated by no less than Herbert Spencer, John Quincy Adams, and George Bernard Shaw (Gardner 1984). In fact, duodecimal still has its advocates, some of whom term it "dozenal."Some aspects of a base-12 system are preserved in the terms dozenand gross applied to the quantities 12 and 144, respectively.The following table gives the duodecimal equivalents of the first few decimal numbers.1111B2119221210221A331311231B44141224205515132521661614262277171527238818162824991917292510A20183026

Goodstein sequence

Given a hereditary representation of a number in base , let be the nonnegative integer which results if we syntactically replace each by (i.e., is a base change operator that 'bumps the base' from up to ). The hereditary representation of 266 in base 2 is(1)(2)so bumping the base from 2 to 3 yields(3)Now repeatedly bump the base and subtract 1,(4)(5)(6)(7)(8)(9)(10)(11)(12)etc.Starting this procedure at an integer gives the Goodstein sequence . Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein's theorem states that is 0 for any and any sufficiently large . Even more amazingly, Paris and Kirby showed in 1982 that Goodstein's theorem is not provable in ordinary Peano arithmetic (Borwein and Bailey 2003, p. 35).

Ternary

The base-3 method of counting in which only the digits 0, 1, and 2 are used. Ternary numbers arise in a number of problems in mathematics, including some problems of weighing. However, according to Knuth (1998), "no substantial application of balanced ternary notation has been made" (balanced ternary uses digits , 0, and 1 instead of 0, 1, and 2).The illustration above shows a graphical representation of the numbers 0 to 25 in ternary, and the following table gives the ternary equivalents of the first few decimal numbers. The concatenation of the ternary digits of the consecutive numbers 0, 1, 2, 3, ... gives (0), (1), (2), (1, 0), (1, 1), (1, 2), (2, 0), ... (OEIS A054635).111110221210221211022211310131112321241114112242205121512025221620161212622272117122271000822182002810019100192012910021010120202301010Ternary digits have the following multiplicationtable.0120000101220211A ternary representation can..

Automatic set

A -automatic set is a set of integers whose base- representations form a regular language, i.e., a language accepted by a finite automaton or state machine. If bases and are incompatible (do not have a common power) and if an -automatic set and -automatic set are both of density 0 over the integers, then it is believed that is finite. However, this problem has not been settled.Some automatic sets, such as the 2-automatic consisting of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (OEIS A048645) have a simple arithmetic expression. However, this is not the case for general -automatic sets.

Normal number

A number is said to be simply normal to base if its base- expansion has each digit appearing with average frequency tending to .A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc. A number that is normal in base- is often called -normal.A number that is -normal for every , 3, ... is said to be absolutely normal (Bailey and Crandall 2003).As stated by Kac (1959), "As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object....It is quite difficult to exhibit a 'normal' number!" (Stoneham 1970).If a real number is -normal, then it is also -normal for and integers (Kuipers..

Absolutely normal

A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).

Digit

The number of digits in an integer is the number of numbers in some base (usually 10) required to represent it. The numbers 1 to 9 are therefore single digits, while the numbers 10 to 99 are double digits. Terms such as "double-digit inflation" are occasionally encountered, although this particular usage has thankfully not been needed in the U.S. for some time. The number of base- digits in a number can be calculated as(1)where is the floor function. For , the formula becomes(2)The number of digits in the number represented in base is given by the Wolfram Language function DigitCount[n, b, d], with DigitCount[n, b] giving a list of the numbers of each digit in . The total number of digits in a number is given by IntegerLength[n, b].The positive integers with distinct base-10 digits are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ... (OEIS A010784). The number of -digit integers is given by(3)(4)(5)(6)where is..

Negadecimal

The negadecimal representation of a number is its representation in base (i.e., base negative 10). It is therefore given by the coefficients in(1)(2)where , 1, ..., 9.The negadecimal digits may be obtained with the WolframLanguage code Negadecimal[0] := {0} Negadecimal[i_] := Rest @ Reverse @ Mod[NestWhileList[(# - Mod[#, 10])/-10&, i, # != 0& ], 10]The following table gives the negadecimal representations for the first few integers(A039723).negadecimalnegadecimalnegadecimal111119121181221219222182331319323183441419424184551519525185661619626186771719727187881819828188991919929189101902018030170The numbers having the same decimal and negadecimal representations are those which are sums of distinct powers of 100: 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, ... (OEIS A051022)...

Negabinary

The negabinary representation of a number is its representation in base (i.e., base negative 2). It is therefore given by the coefficients in(1)(2)where .Conversion of to negabinary can be done using the Wolfram Language code Negabinary[n_Integer] := Module[ {t = (2/3)(4^Floor[Log[4, Abs[n] + 1] + 2] - 1)}, IntegerDigits[BitXor[n + t, t], 2] ]due to D. Librik (Szudzik). The bitwise XOR portion is originally due to Schroeppel (1972), who noted that the sequence of bits in is given by .The following table gives the negabinary representations for the first few integers(OEIS A039724).negabinarynegabinary11111111121101211100311113111014100141001051011510011611010161000071101117100018110001810110911001191011110111102010100If these numbers are interpreted as binary numbers and converted to decimal, their values are 1, 6, 7, 4, 5, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, ... (OEIS A005351). The numbers having the same..

Base

The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].Let the base representation of a number be written(1)(e.g., ). Then, for example, the number 10 is..

Binary plot

A binary plot of an integer sequence is a plot of the binary representations of successive terms where each term is represented as a column of bits with 1s colored black and 0s colored white. The columns are then placed side-by-side to yield an array of colored squares. Several examples are shown above for the positive integers , square numbers , Fibonacci numbers , and binomial coefficients .Binary plots can be extended to rational number sequences by placing the binary representations of numerators on top, and denominators on bottom, as illustrated above for the sequence .Similarly, by using other bases and coloring the base- digits differently, binary plots can be extended to n-ary plots.

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