Decimal

The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, probably because humans normally have a total of ten fingers on their hands.

Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) to represent numbers. These digits are frequently used with a decimal point which indicates the start of a fractional part, and with one of the sign symbols + (plus) or − (minus) to indicate sign. The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc. The position of each digit conveys the multiplier (a power of ten) to be used with that digit. Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius < Lat.) means the unit of ten. However, some cultures do or used to use other numeral systems, including the Tzotzil, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal systems, the Babylonians, who used sexagesimal, and the Yuki, who reportedly used octal. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures. Computers commonly use a different system, binary, internally. For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. For most purposes however, binary values are converted by the computer to the equivalent decimal values for presentation to humans. Nevertheless, sometimes computers do use internal representations which are equivalent to decimal for doing arithmetic. Frequently this arithmetic is done on data in the form of binary-coded decimal, but there are other decimal representations in use (see IEEE 754r). Decimal arithmetic is used in computers so that fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations [http://www2.hursley.ibm.com/decimal/decifaq.html].

Fractional numbers

Decimal fractions

A decimal fraction is a vulgar fraction where the denominator is a power of ten. Decimal fractions can be expressed without a denominator, the decimal point being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. E.g. 8/10, 833/100, 83/1000, 8/10000 and 80/10000 are expressed thus: 0.8, 8.33, 0.083, 0.0008 and 0.008. Numbers which can be expressed in this way are called decimal numbers or regular numbers. The integer and fractional parts of a decimal number are separated by a decimal point. In this article, as in most of the English speaking world, a dot (.) is used. It is usual for a decimal number which is less than one to have a leading zero. Trailing zeroes after the decimal point are not necessary, although in science, engineering and statistics they can be retained to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are mathematically the same number, in engineering 0.080 suggests an error of up to 1 part in a thousand, while 0.08 suggests an error of up to 1 in a hundred (see Significant figures).

Other rational numbers

Any rational number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with recurring decimals. Ten is the product of the first and third prime numbers, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions: :1/2 = 0.5 :1/3 = 0.333333… (with 3 recurring) :1/4 = 0.25 :1/5 = 0.2 :1/6 = 0.166666… (with 6 recurring) :1/8 = 0.125 :1/9 = 0.111111… (with 1 recurring) :1/10 = 0.1 :1/11 = 0.090909… (with 09 recurring) :1/12 = 0.083333… (with 3 recurring) :1/81 = 0.012345679012… (with 012345679 recurring) Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13. That a rational must produce a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only (q-1) possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q-1. For instance to find 3/7 by long division: .4 2 8 5 7 1 4 ... 7 ) 3.0 0 0 0 0 0 0 0 2 8 30/7 = 4 r 2 2 0 1 4 20/7 = 2 r 6 6 0 5 6 60/7 = 8 r 4 4 0 3 5 40/7 = 5 r 5 5 0 4 9 50/7 = 7 r 1 1 0 7 10/7 = 1 r 3 3 0 2 8 30/7 = 4 r 2 (again) 2 0 etc The converse to this observation is that every recurring decimal represents a rational number p/q. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance, :

0.0123123123\cdots = \frac \sum_^\infty 0.001^k = \frac\ \frac = \frac = \frac

The real numbers

Every real number has a (possibly infinite) decimal representation, i.e. it can be written as :

x = \mathop(x) \sum_ a_i\,10^i

where
- sign() is the sign function,
- a_i ∈ for all i ∈ Z, are its decimal digits, equal to zero for all i greater than some number (the common logarithm of |x|). Such a sum always makes sense (i.e. converges), even if there is an infinite number of a_i (with negative indices), which is the case for all reals which are not decimal numbers, according to what precedes. The representation is unique, if one excludes representations that end in a recurring 9. Indeed, consider rational numbers which can be written as p/(2^a5^b) (i.e. the only prime factors in denominator are 2 and 5). In this case there is a terminating decimal representation. For instance 1/1=1, −1/2=−0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999…, −1/2=−0.499999…, etc. Rational numbers p/q with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation. This leaves the irrational numbers. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur. Naturally, the same trichotomy holds for other base-n positional numeral systems:
- Terminating representation: rational where the denominator divides some n^k
- Recurring representation: other rational
- Non-terminating, non-recurring representation: irrational and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.

History

Decimal writers

- c. 3500 - 2500 BC Elamites of Iran possibly use early forms of decimal system. [http://www.chn.ir/english/eshownews.asp?no=1622] [http://www.mpiwg-berlin.mpg.de/Preprints/P183.PDF]
- c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc.).
- c. 2600 BC Indus Valley Civilization, earliest known physical use of decimal fractions in ancient weight system: 1/20, 1/10, 1/5, 1/2. See Ancient Indus Valley weights and measures.
- c. 1400 BC Chinese writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts.
- c. 598–670 Brahmagupta – decimal integers, negative integers, and zero
- c. 790–840 Abu Abdullah Muhammad bin Musa al-Khwarizmi – first to expound on algorism outside India
- c. 920–980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions
- 1548/49–1620 Simon Stevin – author of De Thiende ('the tenth')
- 1561–1613 Bartholemaeus Pitiscus– (possibly) decimal point notation
- 1550–1617 John Napier– decimal logarithms

External links

- [http://www2.hursley.ibm.com/decimal/decifaq.html Decimal arithmetic FAQ]
- Tests: [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1352 Decimal Place Value] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1353&CurriculumID=5 Sums] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=739&CurriculumID=5 Fractions]
- [http://www.mathsisfun.com/worksheets/decimals.php Practice Decimal Arithmetic with Printable Worksheets]
- [http://www.mathsisfun.com/converting-decimals-fractions.html Converting Decimals to Fractions] Category:Elementary arithmetic Category:Numeration Category:Fractions 10 ko:십진법 ja:十進記数法 th:เลขฐานสิบ

Numeral system

A numeral is a symbol or group of symbols that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article attempts to explain the various systems of numerals. See also number names. A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in other bases. Ideally, a numeral system will:
- Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers)
- Give every number represented a unique representation (or at least a standard representation)
- Reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.30999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the natural sciences where differing precision is denoted. Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article.

Types of numeral systems

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol ′ is chosen, for example, then the number seven would be represented by ′′′′′′′. The unary system is normally only useful for small numbers. It has some uses in theoretical computer science. Elias gamma coding is commonly used in data compression; it includes a unary part and a binary part. The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if ′ stands for one, - for ten and + for 100, then the number 304 can be compactly represented as +++ ′′′′ and number 123 as + -- ′′′. The ancient Egyptian system is of this type, and the Roman system is a modification of this idea. More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D′ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted. More elegant is a positional system: again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, borrowed from India, is a positional base 10 system; it is used today throughout the world. Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10). In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-k numeration is also called k-adic notation, not to be confused with p-adic numbers. Bijective base-1 is the same as unary.

History

: See also History of natural numbers and the status of zero. Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, used tallies for gambling with horses, slaves, personal services and trade-goods. The earliest known written tallies appear in the ruins of the Sumerian empire, using clay tablets impressed with a sharp stick and baked. The Sumerians had quite an exotic system based on counts to 60, used in astronomical and other calculations. This system was imported and used by every Mediterranean nation that used astronomy, including the Greeks, Romans and Egyptians. We still use it to count time (minutes per hour), and angle (degrees). In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing. The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman system remained in common use in Europe until positional notation came into common use in the 1500s. The Maya of Central America used a base 20/base 18 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this to do advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus. The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region. Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional records seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle of a place empty of beads. In India, recognizably modern positional numeral systems, passed to the Arabians, probably along with the astronomical tables, was brought to Baghdad by an Indian ambassador around 773. For greater discussion of numeral systems from India, see Arabic numerals and Indian numerals. From India, the thriving trade between Islamic Moguls and Africa carried the concept to Cairo. Arabic mathematicians extended the system to decimal fractions, and al-Khwarizmi wrote an important work about it in the 9th century. The system was introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century. The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications.

Bases used

The base-10 system is the one most commonly used today. It is assumed to have originated because humans have ten fingers. A base-eight system was devised by the Yuki of Northern California, who used the spaces between the fingers to count. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newan, appears to derive from the word for 'new', newo, suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997). (In French, the word neuf still means both 9 and 'new'.) The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base 20 (possibly originating from the number of a person's fingers and toes). Base 60 was used by the Sumerians and their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). 60 is a useful base because it has a large number of factors, including all of the first six counting numbers. Base-12 systems were popular because multiplication is easier in them than in base-10 (addition is just as easy), and because the year has twelve months; we still have a special word for "dozen" and use 12 hours for every night and day. The Nenets language once used a base 9 system, but has since shifted to decimal under the influence of Russian. The word yúq originally meant 9, but took the value 10 on account of Russian influence; so in current Nenets the word for 9 is xasu-yúq, lit. 'Nenets yúq, whereas 10 is simply yúq, but in Eastern dialects also lúca-yúq, lit. 'Russian yúq. Switches (and their electronic successors, built of vacuum tubes, or later of transistors) have only two possible states: "open" and "closed". Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. (In modern transistors, it is more accurate to say that the voltages are high and low instead of 'on' and 'off'). This binary system is the basis for digital computers. It is used to perform integer arithmetic in almost all digital computers, the only exception being the exotic base-3 and base-10 designs that were discarded very early in the history of computing hardware. Note however that a computer does not treat all of its data as integers — some of it may be treated as text and program data. Real numbers (which include numbers other than integers) are usually stored and treated as floating point numbers, which have different rules of arithmetic. The bases that were used in past or used today are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 20, 60.

Positional systems in detail

Also see Positional notation. In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b. For example, in the decimal system (base 10), the numeral 4327 means (4×10³) + (3×10²) + (2×10¹) + (7×10⁰), noting that 10⁰ = 1. In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a₁b^k + a₂b^k-1 + a₃b^k-2 + ... + a_k+1b⁰ and writing the digits a₁a₂a₃ ... a_k+1 in order. The digits are natural numbers between 0 and b-1, inclusive. If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number_base. Unless specified by context, numbers without subscript are considered to be decimal. By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×2¹+ 0×2⁰ +1×2^-1 +1×2^-2 = 2.75. In general, numbers in the base b system are of the form: :

(a_na_...a_1a_0.c_1c_2c_3...)_b = 
\sum_^n a_kb^k + \sum_^\infty c_kb^

The numbers b^k and b^-k are the weights of the corresponding digits. Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3₁₀ = 0.0100110011001...₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...₁₀ can be written down as the unperiodic 11.001001000011111...₂. If b=p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers.

Change of radix

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7:

1020304 / 7 = 145757 r 5
 145757 / 7 =  20822 r 3
  20822 / 7 =   2974 r 4
   2974 / 7 =    424 r 6
    424 / 7 =     60 r 4
     60 / 7 =      8 r 4
      8 / 7 =      1 r 1
      1 / 7 =      0 r 1   => 11446435

E.g., 10110111 base 2 into base 5:

10110111 / 101 = 100100 r 11  (3)
  100100 / 101 =    111 r  1  (1)
     111 / 101 =      1 r 10  (2)
       1 / 101 =      0 r  1  (1)  => 1213

To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately a terminating fraction in one base may not terminate in another. E.g., 0.1A4C base 16 into base 9:

0.1A4C × 9 = 0.ECAC
0.ECAC × 9 = 8.520C
0.520C × 9 = 2.E26C
0.E26C × 9 = 7.F5CC
0.F5CC × 9 = 8.A42C 
0.A42C × 9 = 5.C58C  => 0.082785...

Generalized variable-length integers

More general is using a notation (here written little-endian) like a₀a₁a₂ for a₀ + a₁b₁ + a₂b₁b₂, etc. This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a-z and 0-9, representing 0-25 and 26-35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (1) then a (0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b-9 (1-35), therefore the weight b₁ is 35 instead of 36. Suppose the threshold values for the second and third digit are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence: a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc. Note that unlike a regular base-35 numeral system, we have numbers like 9b where 9 and b each represent 35; yet the representation is unique because ac and aca are not allowed. The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes. The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are nonzero.

Reference

- D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".
- J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.

External links

- [http://www.elfqrin.com/baseconv.html Numeric Base Converter]
- [http://www.kwiznet.com/p/showCurriculum.php?curriculumID=22 Number Sense & Numeration Lessons] Category:Systems ko:기수법 ja:位取り記数法

Base (mathematics)

In mathematics, a base (or radix) is the number of single digits denoting different values in a positional numeral system, including zero. For example, the decimal system, the most common system in use today, uses base ten, hence the maximum number a single digit will ever reach is 9, after this it is necessary to add another digit to achieve a higher number. Sometimes, a subscript notation is used where the base number is written in subscript after the number. For example, a number, say 23, in base 8 would be written as 23₈. This notation will be used in this article.

System

When one says "base b", the b refers to the decimal value of "10" in base b. For example, base 5 means that 10₅ = 5₁₀. The largest digit in a base is therefore one less than the base itself, as after this largest digit, an extra digit must be added to make 10 in that base. Bases work using exponentiation. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of digit to the left the units digit. For example, the number 465 in its respective base (which is clearly at least base 7) is equal to: :

4\times 10^2 + 6\times 10^1 + 5\times 10^0

Numbers that are not integers use places beyond the decimal point. For every point behind the decimal point (and thus the units digit), the power n decreases by 1. For example, the number 2.35 is equal to: :

2\times 10^0 + 3\times 10^ + 5\times 10^

This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values: 241 in base 5: 2 groups of 5² 4 groups of 5 1 group of 1 00000 00000 00000 00000 00000 00000 00000 00000 + + 0 00000 00000 00000 00000 00000 00000 241 in base 8: 2 groups of 8² 4 groups of 8 1 group of 1 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 + + 0 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

Conversion between bases

Bases can be converted between each other by drawing the diagram above and rearranging the objects to conform the new base, for example: 241 in base 5: 2 groups of 5² 4 groups of 5 1 group of 1 00000 00000 00000 00000 00000 00000 00000 00000 + + 0 00000 00000 00000 00000 00000 00000 is equal to 107 in base 8: 1 groups of 8² 0 groups of 8 7 groups of 1 00000000 00000000 0 0 0 00000000 00000000 + + 0 0 00000000 00000000 0 0 0 00000000 00000000 There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step. A number a_na_{n-1_{...a₂a₁a₀ where a₀, a₁... a_n are all digits in a base B (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:

: $\sum_^n \left( a_i\times B^i \right)$

Thus, in the example above:
: $241_5 = 2\times 5^2 + 4\times 5^1 + 1\times 5^0 = 50 + 20 + 1 = 71_$

To convert a decimal number into a base is a slightly more complicated process. One must first find the largest power of the new base that will go into the number. Then, how many whole times the number will go into that power must be found, and the product of the two subtracted from the number. The process is then repeated until one reaches the end.

Thus, to convert 71₁₀ into base 8:
8² goes into 71 once. 8² × 1 = 64; 71 - 64 = 7 units remaining.
8 does not go into 7, therefore still 7 units remaining.
1 goes into 7 seven times.
Therefore $71_ = 1\times 8^2 + 0\times 8^1 + 7\times 8^0 = 107_8$

Applications
The decimal system, base 10, is the base used in everyday life. It is believed that this came about because human beings have ten fingers. However, other civilizations and contexts used different bases.
Historical systems
The Babylonian civilization used a base 60 system. There were not, however, 60 different symbols, as one would expect — each "digit" was represented by a somewhat decimal system, for example, "12 35 1" = 12×60² + 35 ×60 + 1. The Babylonians had their own symbols.

Computing
In computing, the binary (base 2) and hexadecimal (base 16) bases are used. Computers, at the very simplest level, deal only with a series of conventional 1's and 0's, thus it is easier in this sense to deal with powers of two. The hexadecimal system came about as shorthand for binary - every 4 binary digits relates to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B... F.

See also

- Radix

- Numeral system

References

- O'Connor, J. J. and Robertson, E. F. [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html Babylonian numerals]. Retrieved 26 April 2005.

Category:Elementary mathematics

Hand

The hand (med./lat.: manus) is the portion of the arm or anterior limb of a human or other primate, where the appendage terminates. This part of the limb is especially used in grasping and holding. Each hand is a mirror image of the other.

What constitutes a hand?
Although many mammals and other animals have grasping appendages similar in form to a hand, these are scientifically not considered to be so, and have other varying names, including paws. Using the term hand is merely a scientific usage of anthropomorphization, to distinguish the terminations of the front paws from the hind ones. The only true hands appear in the mammalian order of primates. Hands must also feature opposable thumbs, as described later in the text.

Human anatomy of the hand
The human hand consists of a broad palm (metacarpus) with five digits, attached to the forearm by a joint called the wrist (carpus).

Digits
The Five Fingers
Five fingers on the hand are located at the outermost edge of the palm. These four digits can be folded over the palm, this allows for the holding of objects, and furthermore the grasping of small objects. Each finger, starting with the one closest to the thumb, has a colloquial name to distinguish it from the others:

- thumb

- index finger, pointer finger, or forefinger

- middle finger

- ring finger

- little finger or 'pinky'

The thumb
The thumb (connected to the trapezium) is located on one of the sides, parallel to the arm. The thumb can be easily rotated 90º, on a perpendicular level compared to the palm, unlike the fingers which can only be rotated approximately 45º. A reliable way of identifying true hands is from the presence of opposable thumbs. Opposable thumbs are identified by the ability to be brought opposite to the fingers.

opposable thumb]]

Bones
The human hand has at least 27 bones: the carpus or wrist account for 8; the metacarpus or palm contains 5; the remaining 14 are digital bones.

Bones of the wrist
The wrist has eight bones, arranged in two rows of four. These bones fit into a shallow socket formed by the bones of the forearm.

Bones of the palm
The palm has 5 bones, one to each of the 5 digits.

Digital bones
Also called phalanx bones. Human hands contain 14 of them; 2 in the thumb, and 3 in each of the four fingers, called;

- distal phalanx, carrying the nail,

- middle phalanx and

- proximal phalanx.
(The thumb has no middle phalanx).

Sesamoid bones
Sesamoid bones are small ossified nodes embedded in the tendons to provide extra leverage and reduce pressure on the underlying tissue. Many exist around the palm at the bases of the digits, but the exact number varies between different people. The patella is the largest example of a sesamoid bone in the human body.

Muscles and tendons
The movements of the human hand are accomplished by two sets of each of these tissues.
They can be subdivided into two groups: the extrinsic and intrinsic muscle groups.
The extrinsic muscle groups are the long flexors and extensors. They are called extrinsic because the muscle belly is located on the forearm.

Intrinsic hand muscles
The Intrinsic muscle groups are the thenar and hypothenar muscles (thenar referring to the thumb, hypothenar to the small finger), the interosseus muscles (between the metacarpal bones, four dorsally and three volarly) and the lumbrical muscles. These muscles arise from the deep flexor (and are special because they have no bony origin) and insert on the dorsal extensor hood mechanism.

The extrinsic muscles of the hand

The flexors
The fingers have two long flexors, located on the underside of the forearm. They insert by tendons to the phalanges of the fingers. The deep flexor attaches to the distal phalanx, and the superficial flexor attaches to the middle phalanx. The flexors allow for the actual bending of the fingers. The thumb has one long flexor and a short flexor in the thenar muscle group. The human thumb also has other muscles in the thenar group (opponens- and abductor muscle), moving the thumb in opposition, making grasping possible.

The extensors
Located on the back of the forearm and a connected in a more complex way then the flexors to the dorsum of the fingers. The tendons unite with the interosseous and lumbrical muscles to form the extensorhood mechanism. The primary function of the extensors is to straighten out the digits. The thumb has two extensors in the forearm; the tendons of these form the anatomical snuff box. Also, the index finger and the little finger have an extra extensor, used for instance for pointing.

Variation
Some people have more than the usual number of fingers or toes, this is normally caused by a genetic condition called Polydactyly.

Articulation
Polydactyly
Also of note is that the articulation of the human hand is more complex and delicate than that of comparable organs in any other animals. Without this extra articulation, we would not be able to operate a wide variety of tools and devices. The hand can also form a fist, for example in combat, or as a gesture.

See also: Common uses of the word hand in the English language, hand (clock), hand (unit), hand (mechanisms), hand (language).

Common uses in the English language
I know it like the back of my hand - English phrase used to say that the subject knows the matter perfectly, as if it were part of their body, or that they were born with the knowledge. Related: Second hand.

Second hand - Similar to "I know it like the back of my hand," in that it is definitely known by the subject. Similar to something being described as second nature. Not to be confused with second-hand goods, which have already been used before, and are being resold. In the U.S., at least, second hand means indirect--almost the opposite. "She told me walking everyday is good for the brain" indicates second hand knowledge.

A person may also describe somebody as his right hand man, which means that he relies heavily on this person, deriving from the importance and superiority place on the right over the left by many civilizations. Exemplified by phrases such as 'he is my right hand' and 'to be seated at the right hand of the gods when Judgment comes'.

The hand is also an archaic unit of measurement.

Phalanx Bones
measurement

The name Phalanges is commonly given to the bones that form fingers and toes. In primates such as humans and monkeys, the thumb and big toe have two phalanges, while the other fingers and toes consist of three.

The phalanges do not really have individual names but are named after the digit, and their distance from the body. Distal phalanges are at the tips of the fingers and toes, the proximal phalanges are closest to the hand (or foot) and articulate with the metacarpals or metatarsals. Middle phalanges are between the distal and proximal. The thumb and big toe do not have middle phalanges.

The phalanges of the foot correspond with those of the hand. They differ from them in their size (the bodies being much reduced in length) and being laterally compressed.

First Row, The body of each is compressed from side to side, convex above, concave below. The base is concave; and the head presents a trochlear surface for articulation with the second phalanx.

Second Row, The phalanges of the second row are remarkably small and short, but rather broader than those of the first row.

The ungual phalanges, in form, resemble those of the fingers; but they are smaller and are flattened from above downward; each presents a broad base for articulation with the corresponding bone of the second row, and an expanded distal extremity for the support of the nail and end of the toe.

The phalanges are each ossified from two centers: one for the body, and one for the base. The center for the body appears about the tenth week, that for the base between the fourth and tenth years; it joins the body about the eighteenth year.

See also

- handstand

Category:Upper limb anatomy

- [http://www.humanhand.com/ The Human Hand]

ja:手

simple:Hand

Numeral system

A numeral is a symbol or group of symbols that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article attempts to explain the various systems of numerals. See also number names.

A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in other bases.

Ideally, a numeral system will:

- Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers)

- Give every number represented a unique representation (or at least a standard representation)

- Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.30999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the natural sciences where differing precision is denoted.

Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article.

Types of numeral systems

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol ′ is chosen, for example, then the number seven would be represented by ′′′′′′′. The unary system is normally only useful for small numbers. It has some uses in theoretical computer science. Elias gamma coding is commonly used in data compression; it includes a unary part and a binary part.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if ′ stands for one, - for ten and + for 100, then the number 304 can be compactly represented as +++ ′′′′ and number 123 as + -- ′′′. The ancient Egyptian system is of this type, and the Roman system is a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D′ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted.

More elegant is a positional system: again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, borrowed from India, is a positional base 10 system; it is used today throughout the world.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).

In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-k numeration is also called k-adic notation, not to be confused with p-adic numbers. Bijective base-1 is the same as unary.

History
: See also History of natural numbers and the status of zero.

Tallies carved from wood and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, used tallies for gambling with horses, slaves, personal services and trade-goods.

The earliest known written tallies appear in the ruins of the Sumerian empire, using clay tablets impressed with a sharp stick and baked. The Sumerians had quite an exotic system based on counts to 60, used in astronomical and other calculations. This system was imported and used by every Mediterranean nation that used astronomy, including the Greeks, Romans and Egyptians. We still use it to count time (minutes per hour), and angle (degrees).

In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing.

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman system remained in common use in Europe until positional notation came into common use in the 1500s.

The Maya of Central America used a base 20/base 18 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this to do advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.

The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.

Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional records seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle of a place empty of beads.

In India, recognizably modern positional numeral systems, passed to the Arabians, probably along with the astronomical tables, was brought to Baghdad by an Indian ambassador around 773. For greater discussion of numeral systems from India, see Arabic numerals and Indian numerals.

From India, the thriving trade between Islamic Moguls and Africa carried the concept to Cairo. Arabic mathematicians extended the system to decimal fractions, and al-Khwarizmi wrote an important work about it in the 9th century. The system was introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.

The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications.

Bases used

The base-10 system is the one most commonly used today. It is assumed to have originated because humans have ten fingers.

A base-eight system was devised by the Yuki of Northern California, who used the spaces between the fingers to count.
There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newan, appears to derive from the word for 'new', newo, suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997). (In French, the word neuf still means both 9 and 'new'.)

The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base 20 (possibly originating from the number of a person's fingers and toes). Base 60 was used by the Sumerians and their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). 60 is a useful base because it has a large number of factors, including all of the first six counting numbers. Base-12 systems were popular because multiplication is easier in them than in base-10 (addition is just as easy), and because the year has twelve months; we still have a special word for "dozen" and use 12 hours for every night and day.

The Nenets language once used a base 9 system, but has since shifted to decimal under the influence of Russian. The word yúq originally meant 9, but took the value 10 on account of Russian influence; so in current Nenets the word for 9 is xasu-yúq, lit. 'Nenets yúq, whereas 10 is simply yúq, but in Eastern dialects also lúca-yúq, lit. 'Russian yúq.

Switches (and their electronic successors, built of vacuum tubes, or later of transistors) have only two possible states: "open" and "closed". Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. (In modern transistors, it is more accurate to say that the voltages are high and low instead of 'on' and 'off'). This binary system is the basis for digital computers. It is used to perform integer arithmetic in almost all digital computers, the only exception being the exotic base-3 and base-10 designs that were discarded very early in the history of computing hardware. Note however that a computer does not treat all of its data as integers — some of it may be treated as text and program data. Real numbers (which include numbers other than integers) are usually stored and treated as floating point numbers, which have different rules of arithmetic.

The bases that were used in past or used today are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 20, 60.

Positional systems in detail

Also see Positional notation.

In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b.

For example, in the decimal system (base 10), the numeral 4327 means (4×10³) + (3×10²) + (2×10¹) + (7×10⁰), noting that 10⁰ = 1.

In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a₁b^k + a₂b^k-1 + a₃b^k-2 + ... + a_k+1b⁰ and writing the digits a₁a₂a₃ ... a_k+1 in order. The digits are natural numbers between 0 and b-1, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number_base. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×2¹+ 0×2⁰ +1×2^-1 +1×2^-2 = 2.75.

In general, numbers in the base b system are of the form:

: $(a_na_...a_1a_0.c_1c_2c_3...)_b =
\sum_^n a_kb^k + \sum_^\infty c_kb^$

The numbers b^k and b^-k are the weights of the corresponding digits.

Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3₁₀ = 0.0100110011001...₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...₁₀ can be written down as the unperiodic 11.001001000011111...₂.

If b=p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers.

Change of radix
A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7:
1020304 / 7 = 145757 r 5
145757 / 7 = 20822 r 3
20822 / 7 = 2974 r 4
2974 / 7 = 424 r 6
424 / 7 = 60 r 4
60 / 7 = 8 r 4
8 / 7 = 1 r 1
1 / 7 = 0 r 1 => 11446435

E.g., 10110111 base 2 into base 5:
10110111 / 101 = 100100 r 11 (3)
100100 / 101 = 111 r 1 (1)
111 / 101 = 1 r 10 (2)
1 / 101 = 0 r 1 (1) => 1213

To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately a terminating fraction in one base may not terminate in another. E.g., 0.1A4C base 16 into base 9:
0.1A4C × 9 = 0.ECAC
0.ECAC × 9 = 8.520C
0.520C × 9 = 2.E26C
0.E26C × 9 = 7.F5CC
0.F5CC × 9 = 8.A42C
0.A42C × 9 = 5.C58C => 0.082785...

Generalized variable-length integers

More general is using a notation (here written little-endian) like a₀a₁a₂ for a₀ + a₁b₁ + a₂b₁b₂, etc.

This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a-z and 0-9, representing 0-25 and 26-35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (1) then a (0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b-9 (1-35), therefore the weight b₁ is 35 instead of 36. Suppose the threshold values for the second and third digit are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence:

a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc.

Note that unlike a regular base-35 numeral system, we have numbers like 9b where 9 and b each represent 35; yet the representation is unique because ac and aca are not allowed.

The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are nonzero.

See also

- Computer numbering formats

- Billion

- Subtractive notation

- D'ni numerals – a fictional numeral system, from the video game series Myst

- Quipu

- Babylonian numerals – a sexagesimal (base-60) system

- Golden ratio base

Reference

- D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".

- J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.

External links

- [http://www.elfqrin.com/baseconv.html Numeric Base Converter]

- [http://www.kwiznet.com/p/showCurriculum.php?curriculumID=22 Number Sense & Numeration Lessons]

Category:Systems

ko:기수법

ja:位取り記数法

Numerical digit
In mathematics and computer science, a numerical digit is a symbol, e.g. 3, used in numerals (combinations of symbols), e.g. 37, to represent numbers (integers or real numbers) in positional numeral systems. (The name comes from the fact that the 10 fingers correspond to the 10 digits in the common base 10 number system, ie a decimal digit).

Examples of digits include any one of the decimal characters "0" through "9", either of the binary characters "0" or "1", and the digits "0"..."9","A",...,"F" used in the hexadecimal system. In a given number system, if the base (radix) is an integer, the number of needed digits, including zero, is always equal to the absolute value of the base.

(Source: Federal Standard 1037C)

Category:Numeration

ko:숫자

ja:数字

Positional notation
Positional notation or place-value notation is a numeral system in which each position is related to the next by a constant multiplier called the base of that numeral system. Each position may be represented by a unique symbol or by a limited set of symbols. The resultant value of each position is the value of its symbol or symbols multiplied by a power of the base. The total value of a positional number is the total of the resultant values of all positions. The decimal system uses ten unique symbols, whereas the sexagesimal system usually uses a pseudo-decimal system for each position and separates each position from the next by punctuation. Modern computers use binary, octal, and hexadecimal numbers, the latter using decimal numerals (0–9) plus the letters A–F to provide the sixteen possible symbols in each position.

The idea of indicating magnitude by means of position was embodied long ago by the use of the abacus in all its various forms. With an abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system such as Roman Numerals. This approach required no memorization of tables (as does positional notation) and could produce results for all practical purposes very quickly. For four centuries (13th - 16th) there was strong disagreement between those who believed in adopting the positional system and those who wanted to stay with the additive-system-plus-abacus. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern bank checks require a natural language spelling of an amount, as well as the amount itself, to prevent such fraud. The abacus was in widespread use in Japan and other Asian countries until very recent times, when it was replaced by calculators.

The real value of positional notation turned out to be its ability to invite the further study of numbers. Integers, rational numbers, and place-holders (e.g. zero) were long known about, but irrational numbers, infinity, transfinite numbers, and imaginary numbers were all concepts that could only be discovered once the idea of a continuous number line was implied by positional notation.

Decimal system

In the decimal or base 10 number system, each position starting from the right is a higher power of 10. The first position represents 10⁰, the second position 10¹, the third 10², the fourth 10³, and so on.

Fractional values are indicated by a separator, which varies by locale. Usually this separator is a period or full stop, or a comma. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10^-1, the second position 10^-2, and so on for each successive position.

As an example, the number 2674 in a base 10 number system is :

:( 2 × 10³ ) + ( 6 × 10² ) + ( 7 × 10¹ ) + ( 4 × 10⁰ )

or

:( 2 × 1000 ) + ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 )

Sexagesimal system

The sexagesimal or base sixty system was used for the integral and fractional portions of Babylonian numerals, by Hellenistic astronomers using Greek numerals for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds.

Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written 10°25'592331'12 or 10°25^I59^II23^III31^IV12^V.

In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 29;31,50,8,20 days.

Non-positional positions

Each position does not need to be positional itself. Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol), whereas Babylonian numerals used groups of two kinds of wedges representing ones and tens (a narrow vertical wedge ( | ) and an open left pointing wedge (<)) — up to 14 symbols per position (5 tens (<<<<<) and 9 ones ( ||||||||| ) grouped into one or two near squares containing up to three tiers of symbols, or a place holder (\\) for the lack of a position).

A hypothetical Roman numeral positional system would separate each position with punctuation marks but would not necessarily require a symbol for zero. For example, 144 might be I.IV.IV. in decimal notation (medieval Roman numerals were always terminated by a point to show that they were a number). To indicate zero, its position might not be present, for example I.IV.. would mean 140. About 725, Bede or a colleague used N for zero (the initial of the Latin word nulla meaning nothing), so the latter might be I.IV.N.

See also

- Algorism

External links

- [http://www.cut-the-knot.org/binary.shtml Base Converter] at cut-the-knot

- [http://www.cut-the-knot.org/recurrence/conversion.shtml Implementation of Base Conversion] at cut-the-knot

- [http://www.cut-the-knot.org/blue/frac_conv.shtml Conversion of Fractions in Various Bases] at cut-the-knot

- [http://www.cut-the-knot.org/blue/SysTable.shtml Addition and Multiplication Tables in Various Bases] at cut-the-knot

References

- Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.1: Positional Number Systems, pp.195–213.

Category:Numeration
Category:Mathematical notation

DigitA digit is:

- In anatomy, a finger or toe.

- In mathematics and computer science, a numerical digit is a symbol used in the representation of integers or real numbers in positional numeral systems.

- In metrology, a digit (unit) is an ancient unit based on the size of an human finger.

- Digit (magazine) is an Indian information technology magazine.

Unit
The word unit means any of several things:

- Unit of measurement, a fundamental quantity of measurement

- Units (computer program), a popular program that does unit conversion

- Functional unit, a component of a computer system such as the CPU

- Unit of action, a discrete piece of action (or beat) in a theatrical presentation

- Multiple unit, a passenger train whose carriages have their own motors

- United Nations Intelligence Taskforce, a fictional entity in the Doctor Who television series

- Unit, a rock and roll album by the Australian band Regurgitator

- Unit of alcohol, 10 millilitres of pure ethanol in the UK

- In currency, a unit of money (a monetary unit)

- In a 19-inch rack a rack unit is a standard height of 1.75 inches

- In mathematics:

- Unit vector, a vector with length 1

- Unit circle, the circle with radius 1 centered at the origin

- Unit interval, the interval of all real numbers between 0 and 1

- Imaginary unit, i, whose square is -1

- Root of unity, a complex number, a power of which is 1

- Unit (ring theory), an element that is invertible with respect to ring multiplication

- In category theory, there is a natural transformation called the unit from the identity functor to the composition of two adjoint functors, q.v.

- Military units, including:

- Unit 101, an Israeli special operations unit

- Unit 731, a secret unit of the Japanese army

ko:단위

ja:単位

simple:Unit

Tzotzil
The Tzotzil Maya of the central highlands of Chiapas, Mexico are a Native American group, the direct descendants of the Classic Maya. Tzotzil, along with Tzeltal and Ch'ol is descended from the proto-Ch'ol spoken in the late classic period at sites such as Palenque and Yaxchilan. Today, the largest Tzotzil municipalities are Chamula and Zinacantan.

The word "tzotzil" means "people of wool" (tzotz = wool in the Tzotzil language). Tzotzil people make their clothing primarily out of wool. However, according to ancient Maya language, "tzotzil" could also be translated as "bat people", given the association of their culture with this animal in the view of the Mayas.

The Tzotziles were for centuries exploited by Europeans as laborers on coffee and sugar plantations, particularly in the central valleys of the state.

With the collapse of coffee prices in the 1980's, sustainable employment has been hard for many people in the highlands to find. As both population and foreign tourism have risen, the sale of artisan goods has replaced other economic activities. Tzotziles usually sell their products in the nearby cities of San Cristobal de las Casas, Comitán, and Simojovel. Recently, and increasingly, many Maya from the highlands of Chiapas have found migration to other parts of Mexico, and illegal immigration to the United States a way to break away from subsistence farming and abysmal wages.

There still exist some racial/cultural integration issues, especially with white people, mestizos, and westerlized indians (all called "ladinos"). Also, most of the enrollment source for the Zapatista guerrilla are tzotziles.

The Tzotzil dialect of the Mayan language had about 350,000 speakers as of 2002.

References

-

Category:Mayan languages
Category:Languages of Mexico
Category:Indigenous peoples of Mexico

Vigesimal

The vigesimal or base-twenty numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). Twenty is the sum of all fingers and toes on human hands and feet, and is the product of five and four.

Places

In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the usual decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A, to write nineteen as J, and the numbers between with the corresponding letters of the alphabet. (This is an extension of the common practice of writing hexadecimal numerals over 9 with the letters "A-F".) Another method skips over the letter "I", in order to avoid confusion between I as eighteen and 1 (one), so that the number eighteen is written as J, and nineteen is written as K. The number twenty is written as "10".

According to this notation, 20 means forty (= two times twenty), DA means two hundred seventy (= thirteen
and ten-twentieth times twenty), 100 means four hundred (= twenty times twenty). In the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example, 10 means ten, 20 means twenty.

Usage
In many languages, especially in Europe, 20 is a base, at least with respect to the linguistic structure of the names of certain numbers (though a thoroughgoing consistent vigesimal system, based on the powers 20, 400, 8000 etc., is not generally used):

- Twenty () is used as a base number in the French language names of numbers from 70-99. So for example, means 4 times 20, i.e. is the French word for 80, and (literally "sixty-fifteen") means 75. Controversial Japanese politician Shintaro Ishihara used this to accuse the French of being unable to count.

- Twenty () is used as a base number in the Danish language: (short for ) means 3 times 20, i.e. 60; (short for ) means 4 times 20 i.e. 80. means (3 － 1/2) times 20, i.e. 50; means (4 － 1/2) times 20, i.e. 70; and means (5 － 1/2) times 20, i.e. 90.

- Twenty () is used as a base number in the Welsh language, although in the latter part of the twentieth century a decimal counting system came to be preferred, with the vigesimal system becoming 'traditional'. means 2 times 20 i.e. 40, means 3 times 20 i.e. 60. Prior to the currency decimalisation in 1971, (6 times 20 paper) was the nickname for the 10 shilling (= 120 pence) note.

- Twenty () is used as a base number in the Georgian language. For example, 31 () literally means, twenty-and-eleven. 67 () is said as, “three-twenty-and-seven”.

- Twenty () is used as a base number in the Albanian language. The word for 40 () means two times 20.

- Twenty () is used as a base number in the Basque language for numbers up to 100 (). The words for 40 (), 60 () and 80 () mean "two-score", "three-score" and "four-score", respectively. The number 75 is called , lit. "three-score-and ten-five".

- Twenty was the base of the Maya and Aztec number systems. The Maya used the following names for the powers of twenty: (20), (20² = 400), (20³ = 8,000), (20⁴ = 160,000), (20⁵ = 3,200,000) and (20⁶ = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: (1 × 20), (1 × 400), (1 × 8,000), (1 × 20 × 8,000 = 160,000), (1 × 400 × 8,000 = 3,200,000) and (1 × 20 × 400 × 8,000 = 64,000,000). Note that the prefix at the beginning means "one" (as in "one hundred" and "one thousand") and is replaced with the corresponding number to get the names of other multiples of the power. For example, (2) × (20) = (40), (2) × (400) = (800). Note also that the in (and ) and the in are grammatical noun suffixes that are appended only at the end of the word; thus , and compound together as (instead of
- ). (See also Nahuatl language.)

- In the old British currency system, there were 20 shillings in a pound.

- In English, counting by the score has been used historically, as in the famous opening of the Gettysburg Address "Four score and seven years ago…", meaning eighty-seven (87) years ago. This method has fallen into disuse, however.

- Among multiples of 10, 20 is described in a special way in some languages. For example, the Spanish words (30) and (40) consist of " (10 times)", " (10 times)", but the word (20) is not synchronically connected to any word meaning "two" (although, diachronically, it is). Similarly, in Semitic languages such as Arabic and Hebrew, the numbers 30, 40 ... 90 are expressed by morphologically plural forms of the words for the numbers 3, 4 ... 9, but the number 20 is expressed by a morphologically plural form of the word for 10.

- In many languages, the names of the two-digit numbers from 11 to 19 consist of one word, but the names of the two-digit numbers from 21 on consist of two words. So for example, the English words eleven (11), twelve (12), thirteen (13) etc., as opposed to twenty-one (21), twenty-two (22), twenty-three (23), etc. In a number of other languages (such as Hebrew), the names of the numbers from 11-19 contain two words, but one of these words is a special "teen" form which is different from the ordinary form of the word for the number 10, and may in fact be only found in these names of the numbers 11-19.

- In East Asia, the Ainu language also uses a counting system that is based around the number 20. “” is 20, “” (ten more until two twenties) is 30, “” (two twenties) is 40, “” (five twenties) is 100. Subtraction is also heavily used, e.g. “” (one more until ten) is 9.

Origin in Europe
According to German linguist Theo Vennemann, the vigesimal system in Europe is of Basque (Vasconic) origin and spread from the Vasconic languages to other European tongues, such as many Celtic languages, French and Danish.

However according to Menninger the vigesimal system originated with the Normans and spread through them to Western Europe.

Further reading

- Karl Menninger: Number words and number symbols: a cultural history of numbers; translated by Paul Broneer from the revised German edition. Cambridge, Mass.: M.I.T. Press, 1969 (also available in paperback: New York: Dover, 1992 ISBN 0-486-27096-3)

- Levi Leonard Conant: The Number Concept: Its Origin and Development; New York, New York: MacMillon & Co, 1931. [http://www.gutenberg.org/etext/16449 Project Gutenberg EBook]

See also

- quinary

- decimal

- duodecimal

- sexagesimal

Notes

# The diachronic view is like this. < , the IE etymology of which ([http://starling.rinet.ru/cgi-bin/response.cgi?root=config&morpho=0&basename=%5Cdata%5Cie%5Cpiet&first=1&text_proto=&method_proto=substring&text_meaning=&method_meaning=substring&text_rusmean=&method_rusmean=substring&text_hitt=&method_hitt=substring&text_ind=&method_ind=substring&text_avest=&method_avest=substring&text_iran=&method_iran=substring&text_arm=&method_arm=substring&text_greek=&method_greek=substring&text_slav=&method_slav=substring&text_balt=&method_balt=substring&text_germ=&method_germ=substring&text_lat=v%C4%ABgint%C4%AB&method_lat=substring&text_ital=&method_ital=substring&text_celt=&method_celt=substring&text_alb=&method_alb=substring&text_tokh=&method_tokh=substring&text_refer=&method_refer=substring&text_comment=&method_comment=substring&text_any=&method_any=substring&sort=proto view]) connects it to the roots meaning [http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=/data/ie/pokorny&text_number=+328&root=config `2'] and [http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=/data/ie/pokorny&text_number=+369&root=config `10']. (The [http://starling.rinet.ru/cgi-bin/main.cgi?flags=eygtnnl etymological databases] of the [http://starling.rinet.ru/main.html Tower of Babel] project are referred here.)

Category:Numeration
20

ja:二十進記数法

Nigeria
The Federal Republic of Nigeria is a country in West Africa. It is the most populous country in Africa. Nigeria re-achieved democracy in 1999 after a long sixteen years interruption by corrupt and brutal series of military dictators and counter coups. Nigeria borders Republic of Benin in the west, Chad and Cameroon in the east, Niger in the north and the Gulf of Guinea in the south. Major cities include the capital Abuja, the former capital Lagos, Ibadan, Osogbo, Port Harcourt, Enugu, Kano, Kaduna, Onitsha, Jos, Ilorin, Maiduguri, Bauchi, Sokoto and Benin City. The country's name first appeared in print in the Times of London in 1897 and was suggested by the papers colonial editor Flora Shaw who would later marry Frederick Lugard, the first Govenor General of the Amalgamated Nigeria. The name comes from a combination of the words "Niger" (the country's longest river) and "Area." Its adjective form is Nigerian , which should not be confused with Nigerien for Niger.
History

The first known civilization in Nigeria was that of the Nok. The Nok were an iron age people existing from 500 BC until about 200 AD on the Jos plateau in north-eastern Nigeria.

The Kanem-Bornu Empire near Lake Chad dominated northern Nigeria for over 600 years, prospering as a terminal of north-south trade between North African Berbers and forest people. In the early 19th century, Usman dan Fodio brought most areas in the north under the loose control of an Islamic empire centered at Sokoto.

The kingdoms of Ife and Oyo in the southwest and Benin in the south developed elaborate systems of political organization in the 15th, 16th and 17th centuries. Ife and Benin are noted for their prized artistic works in ivory, wood, bronze, and brass.

In the southeast, the populous village-networks of the Igbo and other acephalous groups like the Ibibio were governed by indigenous African notions of egalitarianism and democracy. Some of the oldest artwork found in West Africa was recovered in this region, with the Igbo-Ukwu bronze sculptures being among the most famous.

In the 17th through 19th centuries, European traders established coastal ports for the increasing traffic in slaves destined for the American continent. Commodity trade replaced slave trade in the 19th century.

The Royal Niger Company was chartered by the British government in 1886. Northern and Southern Nigeria became British protectorates in 1901 and were amalgamated into a single colony in 1914. In response to the growth of Nigerian nationalism following World War II, the British moved the colony towards self-government on a federal basis.

Nigeria won full independence in 1960, as a federation of three regions, each retaining a substantial measure of self-government. At the time of Nigeria's first elections in 1959, there were a number of prominent parties - Nnamdi Azikiwe's National Council of Nigerian and the Cameroons (NCNC) which had control of the Eastern Region, Ahmadu Bello's Northern People's Congress (NPC), which had control of the Northern Region and Obafemi Awolowo's Action Group (AG) which had control of the Western Region.

When no party won a majority during the 1959 elections, the NPC combined with the NCNC to form a government, and when independence arrived in 1960, Abubakar Tafawa Balewa was made the Prime Minister, and Nnamdi Azikiwe was made the Governor General.

As with much of Nigerian history, severe conflicts developed within the ruling coalition. In 1962, part of the Action Group split off to form the Nigerian National Democratic Party (NNDP), led by S.I. Akintola. In 1963, the Mid-Western Region was formed from part of the Western Region. When Nigeria became a Republic in 1963, Nnamdi Azikiwe was made the President of the Federal Republic. However, in 1964, a great controversy broke out, over the 1963 population census, with the NCNC claiming that there was an overestimatation of the number of people in the Northern Region, thus giving the north a greater representation in the federal parliament.

In 1966, two successive coups by different groups of army officers brought the country under military rule. In January of that year, a number of junior army officers staged a coup d'etat to overthrow the government, in the process killing Balewa, Bello, Akintola and some senior officers. Johnson Aguiyi-Ironsi who successfully stopped the coup, was put in charge of the military government which was to be the first of many. Despite the fact that this coup was tremendously violent, the new government did promise a progressive agenda - a return to civilian rule determined by elections and vowed to stamp out corruption and stop violence, and this particularly appealed to the youth. Furthermore, Aguiyi-Ironsi tried to restore discipline within the army. He suspended the regional constitution with its different regions, dissolved all legislative bodies, banned political parties, imprisoned Awolowo, and formed a Federal Military Government with the aim of centralising governance. A decree was issued, that March, to abolish the federation, and unify the federal and regional civil servants. Many accused Aguiyi-Ironsi of favouring the Igbos over other ethnic groups and the fact that the military government did not prosecute the officers that killed the northern leaders stirred further rage. Though Aguiyi-Ironsi had some concessions like protecting the northerners from southern competition in the civil service, many northerners felt like the coup was a plot to make the Igbo's dominant in Nigeria. Fighting broke out for a while between the northerners and the Igbo, and in July of the same year, northern officers staged another coup, killing Aguiyi-Ironsi and many other Igbo officials. The Muslim officers chose Yakubu Gowon (who was a Christian) as the new ruler. Gowon had not actually been involved in the coup, but they felt he would be a compromising candidate to head the Federal Military Government. His first steps included restoring Federalism, and releasing Awolowo from prison.

Gowon vowed to start Nigeria along the road to civilian government. However, now the Igbos were becoming more and more afraid of their position in Nigeria. In 1967, when Gowon moved to split the 4 existing regions into 12 states, Chukwuemeka Ojukwu, the leader of the Eastern Region refused to accept this, and declared that the Eastern Region would become its own independent republic, named Biafra. This was not accepted, and in June 1967, a civil war broke out between Biafra and the remainder of Nigeria.

Following the creation of Biafra, war broke out between the Federal Government and the Igbo dominated eastern region. Under Brigadiers Adekunle, Obasanjo and Murtala Mohammed, a systematic battle plan that comprised saturated air bombings and starvation forced the Biafran rebels to capitulate. On 15th January 1970, left with the choice of surrender and the total destruction of the Biafran populace, Philip Effiong, Chief of Staff of the rebel army accepted the terms of surrender before Yakubu Gowon, Head of the Northern dominated federal government.

In 1974, Gowon broke his promise to return the nation to civilian rule, and in July of 1975, there was yet another military coup, the first of many bloodless coups. This brought the hugely popular Murtala Ramat Mohammed to power. As his predecessors had done, Murtala Mohammed promised to lead Nigeria back into civilian rule. In February of 1976, there was an attempted coup by Buka Dimka, and though it was unsuccessful, Muhammed was killed. So, Olusegun Obasanjo was chosen to take his place as the new ruler, and promised to continue what Muhammed had started. During his term, he raised University fees, and this led to student riots (which have also become quite common it seems). The government then banned student organizations, restricted public opposition to the regime, controlled union activity, and nationalized land. Controversy trailed his indigenization of foreign businesses perceived to be much to the advantage of his own Yoruba people who were the larger population in the then capital Lagos and the increased oil industry regulation. However, in 1978, Obasanjo did set up a new constitution, one that would return the country to the much awaited state of civilian rule. Elections were finally held in 1979, bringing Shehu Shagari into office as the new President of Nigeria.

While Shagari was able to serve his entire term and was, in fact the victor of the 1983 elections, many people believed the elections were rigged and the rightful leader was Obafemi Awolowo. This set the stage for yet another coup, this time on December 31, 1983. The new military government, under Muhammadu Buhari was welcomed at the time, because many felt that the nation had further deteriorated into more shameless corruption and economic mismanagement, under the supposedly democratically elected government of Shehu Shagari. Buhari set out to try to revive the economy, and this took priority over everything else, including returning the country to civilian rule. He also took security of the government as a high priority, restricted freedom of the press, suppressed criticism of the government, and outlawed many organizations. Moreover, he declared a "War Against Indiscipline" to deal with such aspects as public behavior, sanitation, public appearance, corruption, smuggling, and patriotism. He also took many other measures of austerity that made it difficult for some companies to run, and this eventually led to high inflation and thus a much higher cost of living.

Yet another bloodless coup took place on August 27, 1985. This time Ibrahim Babangida (Buhari's chief of army staff before the coup) was named the ruler. Babangida claimed that Buhari's regime was insensitive to the feelings of the Nigerian masses, especially with regards to the restrictions imposed on the press. He started his rule claiming to be a human rights activist, but this image faded with time. Though he released some of the politicians that Buhari incarcerated, he also hounded opposition interest groups, and detained many radical people for various offenses, and even had a decree to facilitate some oppressive acts. As concerns his economic policy, Babangida introduced market reforms, freeing exchange and interest rates, and this led to a sharp drop in the value of the Nigerian currency, while raising lending rates to more than 40 percent.

In April of 1986, there was another attempted coup by Mamman Vatsa, and him and his followers were executed. On April 22, 1990, there was yet another attempted coup by Gideon Orkar that failed, but almost killed Babangida, whose bedroom had been bombed. Unlike previous coups and attempted coups, this coup was believed to have been heavily funded by civilians, suggesting that they were willing to have another military ruler over Babangida.

As per a new constitution that was drafted in 1990, the country was to return to civilian rule in 1992. As the date approached, there were many suspicions that this promise was not going to be kept. Pressure started mounting on the mi}}