Dictionary Definition
radix n : (numeration system) the positive
integer that is equivalent to one in the next higher counting
place; "10 is the radix of the decimal system" [syn: base] [also: radices (pl)]
User Contributed Dictionary
English
Noun
radix A primitive word, from which other words spring.
Translations
 French: racine
Latin
Noun
Derived terms
 rót ()
Extensive Definition
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. 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
nonstandard 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 . Thus,
binary numbers have digits ; decimal numbers have digits ; 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.
For example, 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:
 4\times b^2 + 6\times b^1 + 5\times b^0
If the number 465 was in base 10, then it would
equal:
 4\times 10^2 + 6\times 10^1 + 5\times 10^0 = 4\times 100 + 6\times 10 + 5\times 1 = 465
If however, the number were in base 7, then it
would equal:
 4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 49 + 6\times 7 + 5\times 1 = 243
10b = b for any base b, since 10b = 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
onedigit 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:
 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 (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 nonintegers 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:
 1\times 3^ +
 1\times 3^ + 2\times 3^ +
 1\times 3^ + 1\times 3^ + 2\times 3^ +
 1\times 3^ + 1\times 3^ + 1\times 3^ + 2\times 3^ +
 1\times 3^ + 1\times 3^ + 1\times 3^ + 1\times 3^ + 2\times 3^ + ...
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:
 2.42\overline_5 = 2.42314314314314314..._5
For base 10 it is called a recurring
decimal or repeating decimal.
An irrational
number has an infinite nonrepeating 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_ = 0.3333333..._
 0.\overline_2 = 0.010101..._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 = 3.46\overline0_7
 2. The last nonzero 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 = 3.45\overline6_7
 1_ = 0.\overline9_
 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 ooooois 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
nondecimal bases without using this intermediate step.
A number anan1...a2a1a0 where a0, a1... an 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 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 binarylike counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon, ukasar, ukasarurapon, ukasarukasar, ukasarukasarurapon, ukasarukasarukasar.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 (basetwenty)
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.
References
 O'Connor, J. J. and Robertson, E. F. Babylonian numerals. Retrieved 26 April 2005.
External links
radix in French: Base (arithmétique)
radix in Hebrew: בסיס (לשיטת ספירה)
radix in Latvian: bāze (matemātika)
radix in Dutch: Grondtal
radix in Thai: Radix