Positional notation

 ( Mathematical Notation ) 

241 in base 5:
   2 groups of 52 (25)           4 groups of 5          1 group of 1
   ooooo    ooooo
   ooooo    ooooo                ooooo   ooooo
   ooooo    ooooo         +                         +         o
   ooooo    ooooo                ooooo   ooooo
   ooooo    ooooo
     

241 in base 8:
   2 groups of 82 (64)          4 groups of 8          1 group of 1
 oooooooo  oooooooo
 oooooooo  oooooooo
 oooooooo  oooooooo         oooooooo   oooooooo
 oooooooo  oooooooo    +                            +        o
 oooooooo  oooooooo
 oooooooo  oooooooo         oooooooo   oooooooo
 oooooooo  oooooooo
 oooooooo  oooooooo
     

0xA10B/10 = Q: 0x101A R: 7 (ones place)
0x101A/10 = Q: 0x19C  R: 2 (tens place)
 0x19C/10 = Q: 0x29   R: 2 (hundreds place)
  0x29/10 = Q: 0x4    R: 1  ...
                         4
     

0b11111001/10 = Q: 0b11000 R: 0b1001 (0b1001 = "9" for ones place)
   0b11000/10 = Q: 0b10    R: 0b100  (0b100 =  "4" for tens)
      0b10/10 = Q: 0b0     R: 0b10   (0b10 =   "2" for hundreds)
     

Convert 0xA10B to 41227
 A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0)
     

 Lookup table:
  0x0 = 0
  0x1 = 1
  ...
  0x9 = 9
  0xA = 10
  0xB = 11
  0xC = 12
  0xD = 13
  0xE = 14
  0xF = 15
 Therefore 0xA10B's decimal digits are 10, 1, 0, and 11.
     

 Lay out the digits out like this. The most significant digit (10) is "dropped":
  10 1   0    11 <- Digits of 0xA10B
     

  ---------------
  10
 Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add:
  10 1   0    11
     160
  ---------------
  10 161
     

 Repeat until the final addition is performed:
  10 1   0    11
     160 2576 41216
  ---------------
  10 161 2576 41227
     

 and that is 41227 in decimal.
     

Convert 0b11111001 to 249
 Lookup table:
  0b0 = 0
  0b1 = 1
     

Result:
 1  1  1  1  1  0  0   1 <- Digits of 0b11111001
    2  6  14 30 62 124 248
 -------------------------
 1  3  7  15 31 62 124 249
     

$\backslash frac\{\backslash N\_0\}\{b^\{\backslash N\_0\}\}\; :=\; \backslash left\backslash \{mb^\{-\backslash nu\}\backslash mid\; m\backslash in\; \backslash N\_0\; \backslash wedge\; \backslash nu\backslash in\; \backslash N\_0\; \backslash right\backslash \}\; .$

$\backslash Z\_S\; :=\; \backslash left\backslash \{x\; \backslash in\; \backslash Q\; \backslash left\; |\; \backslash ,\; \backslash exists\; \backslash mu\_i\; \backslash in\; \backslash Z\; :\; x\; \backslash prod\_\{i=1\}^n\; \{p\_i\}^\{\backslash mu\_i\}\; \backslash in\; \backslash Z\; \backslash right\; .\; \backslash right\backslash \}\; =\; b^\{\backslash Z\}\; \backslash ,\; \backslash Z\; =\; \{\backslash langle\; S\backslash rangle\}^\{-1\}\backslash Z$

The denominator of an element of $\backslash Z\_S$ contains if reduced to lowest terms only prime factors out of $S$. This ring of all terminating fractions to base $b$ is Dense set in the field of $\backslash Q$. Its completion for the usual (Archimedean) metric is the same as for $\backslash Q$, namely the real numbers $\backslash R$. So, if $S\; =\; \backslash \{\; p\backslash \}$ then $\backslash Z\_\{\backslash \{\; p\backslash \}\}$ has not to be confused with $\backslash Z\_\{(p)\}$, the discrete valuation ring for the prime number $p$, which is equal to $\backslash Z\_\{T\}$ with $T\; =\; \backslash mathbb\; P\; \backslash setminus\; \backslash \{\; p\backslash \}$.

If $b$ divides $c$, we have $b^\{\backslash Z\}\; \backslash ,\; \backslash Z\; \backslash subseteq\; c^\{\backslash Z\}\; \backslash ,\; \backslash Z.$

$\backslash begin\{array\}\{l\}$

Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a vinculum across the repeating block:

$2.42\backslash overline\{314\}\_5\; =\; 2.42314314314314314\backslash dots\_5$

This is the repeating decimal notation (to which there does not exist a single universally accepted notation or phrasing). For base 10 it is called a repeating decimal or recurring decimal.

An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:

$0.1\_3$

$0.\backslash overline3\_\{10\}\; =\; 0.3333333\backslash dots\_\{10\}$

: or, with the base implied:

: $0.\backslash overline3\; =\; 0.3333333\backslash dots$ (see also 0.999...)

$0.\backslash overline\{01\}\_2\; =\; 0.010101\backslash dots\_2$

$0.2\_6$

For integers p and q with gcd ( p, q) = 1, the fraction p/ q has a finite representation in base b if and only if each prime factor of q is also a prime factor of b.

For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:

1. A finite or infinite number of zeroes can be appended:
2. : $3.46\_7\; =\; 3.460\_7\; =\; 3.460000\_7\; =\; 3.46\backslash overline0\_7$
3. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
4. : $3.46\_7\; =\; 3.45\backslash overline6\_7$
5. : $1\_\{10\}\; =\; 0.\backslash overline9\_\{10\}\backslash qquad$ (see also 0.999...)
6. : $220\_5\; =\; 214.\backslash overline4\_5$

Examples are the non-solvable nth root

$y\; =\; \backslash sqrtn\{x\}$

$\backslash pi,e$

Decimal values are indicated by a separator, which can vary in different locations. 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 (0.1), the second position 10⁻² (0.01), and so on for each successive position.

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

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

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

Modern time separates each position by a colon or a prime symbol. 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 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 , they would have written or .

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.

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, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.

The octal numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.

Hexadecimal, decimal, octal, and a wide variety of other bases have been used for binary-to-text encoding, implementations of arbitrary-precision arithmetic, and other applications.

For a list of bases and their applications, see list of numeral systems.

The Maya numerals and other civilizations of pre-Columbian Mesoamerica used base-20 (vigesimal), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa.

Remnants of a Gaulish language base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty and five"), while seventy-five is soixante-quinze (literally, "sixty and fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is quatre-vingt-deux (literally, four twentys and two), while ninety-two is quatre-vingt-douze (literally, four twentys and twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties and thirteen, and so on.

In English the same base-20 counting appears in the use of "scores". Although mostly historical, it is occasionally used colloquially. Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago".

The Irish language also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.

The Welsh language continues to use a vigesimal counting system, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.

The Inuit languages use a base-20 counting system. Students from Kaktovik, Alaska invented a base-20 numeral system in 1994

Danish numerals display a similar vigesimal structure.

The Māori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama-hokotahi, referring to a great warrior ("the one man equal to 20").

The binary system was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).

A number of Australian Aboriginal languages employ binary or binary-like counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.

North and Central American natives used base-4 (quaternary) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.

A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.

A base-8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.. 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, newm, is suggested by some 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) Encyclopedia of Indo-European Culture

Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for five is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda languages of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region.

The Telefol language, spoken in Papua New Guinea, is notable for possessing a base-27 numeral system.

Balanced ternaryKnuth, pages 195–213 uses a base of 3 but the digit set is instead of {0,1,2}. The "" has an equivalent value of −1. The negation of a number is easily formed by switching the on the 1s. This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ..., 3 ⁿ known units can be used to determine any unknown weight up to 1 + 3 + ... + 3 ⁿ units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with , with 1 if used on the empty pan, and with 0 if not used. If an unknown weight W is balanced with 3 (3¹) on its pan and 1 and 27 (3⁰ and 3³) on the other, then its weight in decimal is 25 or 101 in balanced base-3.

The factorial number system uses a varying radix, giving as place values; they are related to Chinese remainder theorem and residue number system enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.

• List of numeral systems
• Positional numeral systems

Related topics:

• Algorism
• Hindu–Arabic numeral system
• Mixed radix
• Non-standard positional numeral systems
• Scientific notation

Other:

• Significant figures

• Donald Knuth (1997). 9780201896848, Addison-Wesley. ISBN 9780201896848
• George Ifrah (2025). 9780471375685, Wiley. . ISBN 9780471375685
• Alfred Kroeber (1976). 9780486233680, Courier Dover Publications. . ISBN 9780486233680

• Accurate Base Conversion
• The Development of Hindu Arabic and Traditional Chinese Arithmetics
• Implementation of Base Conversion at cut-the-knot
• Learn to count other bases on your fingers
• Online Arbitrary Precision Base Converter