In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is 10, because it uses the 10 digits from 0 through 9.

The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the radix they use. The radix itself is almost always expressed in decimal notation, because if the radix were expressed instead in the numeral system it was trying to describe, it would always be written as "10"—the radix of the binary system, two, is expressed in binary as "10"; the radix of the decimal system, ten, is expressed in decimal as "10"; the radix of the hexadecimal system, sixteen, is expressed in hexadecimal as "10", and so forth. The radix is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.

In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits deviates from the above.

Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, $23\_8$ indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.

System

When describing radix in mathematical notation, the letter b is generally used as a symbol for this concept, so, for a binary system, b equals 2. Another common way of expressing the radix is writing it as a decimal subscript after the number that is being represented. 1111011₂ implies that the number 1111011 is a base 2 number, equal to 123₁₀ (a decimal notation representation), 173₈ (octal) and 7B₁₆ (hexadecimal). When using the written abbreviations of number bases, the radix is not printed: Binary 1111011 is the same as 1111011₂.

The radix b may also be indicated by the phrase "base b". So binary numbers (radix 2) have base 2; octal numbers (radix 8) have base 8; decimal numbers (radix 10) have base 10; and so on.

Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 52₂, 2₂, 1A₉. (In all cases, one or more digits is not in the set of allowed digits for the given 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 other digits between the current digit and the decimal point. If the current digit is on the left hand side of the decimal point (i.e., it is greater than or equal to 1) then n is positive; if the digit is on the right hand side of the decimal point (i.e., it is fractional) then n is negative.

As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:

$4times\; b^2\; +\; 6times\; b^1\; +\; 5times\; b^0$

If the number 465 was in base 10, then it would equal:

$4times\; 10^2\; +\; 6times\; 10^1\; +\; 5times\; 10^0\; =\; 4times\; 100\; +\; 6times\; 10\; +\; 5times\; 1\; =\; 465$

(465₁₀ = 465₁₀)

If however, the number were in base 7, then it would equal:

$4times\; 7^2\; +\; 6times\; 7^1\; +\; 5times\; 7^0\; =\; 4times\; 49\; +\; 6times\; 7\; +\; 5times\; 1\; =\; 243$

(465₇ = 243₁₀)

10_b = b for any base b, since 10_b = 1×b¹ + 0×b⁰. For example 10₂ = 2; 10₃ = 3; 10₁₆ = 16₁₀. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.

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

$2times\; 10^0\; +\; 3times\; 10^\{-1\}\; +\; 5times\; 10^\{-2\}$

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 (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

Infinite representations

The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example 1.12112111211112 ... base 3 represents the sum of the infinite series:

$1times\; 3^\{0,,,\}\; +\; \{\}$

$1times\; 3^\{-1,,\}\; +\; 2times\; 3^\{-2,,,\}\; +\; \{\}$

$1times\; 3^\{-3,,\}\; +\; 1times\; 3^\{-4,,,\}\; +\; 2times\; 3^\{-5,,,\}\; +\; \{\}$

$1times\; 3^\{-6,,\}\; +\; 1times\; 3^\{-7,,,\}\; +\; 1times\; 3^\{-8,,,\}\; +\; 2times\; 3^\{-9,,,\}\; +\; \{\}$

$1times\; 3^\{-10\}\; +\; 1times\; 3^\{-11\}\; +\; 1times\; 3^\{-12\}\; +\; 1times\; 3^\{-13\}\; +\; 2times\; 3^\{-14\}\; +\; cdots$

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 bar across the repeating block:

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

For base 10 it is called a recurring decimal or repeating 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:

$0.1\_3,$

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

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

$0.2\_6,$

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:

$3.46\_7\; =\; 3.460\_7\; =\; 3.460000\_7\; =\; 3.46overline0\_7$

2. 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):

$3.46\_7\; =\; 3.45overline6\_7$

$1\_\{10\}\; =\; 0.overline9\_\{10\}$

$220\_5\; =\; 214.overline4\_5$

Relationship between real numbers and their representations

The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.

Conversion among 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

   ooooo    ooooo

   ooooo    ooooo           ooooo   ooooo

   ooooo    ooooo     +                        +         o

   ooooo    ooooo           ooooo   ooooo

   ooooo    ooooo

is equal to 107 in base 8:

    1 group of 8²           0 groups of 8          7 groups of 1

      oooooooo

      oooooooo                                        o     o

      oooooooo

      oooooooo        +                        +    o    o    o

      oooooooo

      oooooooo                                        o     o

      oooooooo

      oooooooo

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:

$sum\_\{i=0\}^n\; left(a\_itimes\; b^i\; right)$

Thus, in the example above:

$241\_5\; =\; 2times\; 5^2\; +\; 4times\; 5^1\; +\; 1times\; 5^0\; =\; 50\; +\; 20\; +\; 1\; =\; 71\_\{10\}$

To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.

The most common example is that of changing from Decimal to Binary.

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 (including two thumbs). 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 modified decimal system, for example, "12 35 1" = 12×60² + 35×60 + 1. The Babylonians had their own number symbols.

Other bases in human language

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.

Various traditional systems of measurement use duodecimal reckoning (base twelve), which in English is represented by terms such as dozen (12) and gross (144 = 12 x 12), and measurements such as foot (12 inches).

Certain European languages including Basque, French and Danish incorporate elements of a vigesimal (base-twenty) counting system. The Maya and Aztecs in Mesoamerica used vigesimal, as do the Ainu in East Asia.

Computing

In computing, the binary (base 2) and hexadecimal (base 16) bases are used. Computers, at the very simplest level, deal only with sequences of conventional 1s and 0s, 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, C, D, E and F (sometimes a, b, c, d, e, f).

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 binary digits relate to one and only one octal digit.

See also

• Mixed radix
• Negative base
• Radix point
• Radix sort
• Decimal superbase

References

• O'Connor, J. J. and Robertson, E. F. Babylonian numerals Retrieved 26 April 2005.

External links

• MathWorld entry on base