Radix

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In mathematical numeral systems, the base or radix is usually the number of various unique digits, including zero, that a positional numeral system uses to represent numbers. 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 which it is necessary to add another digit to achieve a higher number.

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, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit.

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.

1 System
2 Infinite representations
3 Relationship between real numbers and their representations
4 Conversion among bases
5 Applications
6 See also
7 References
8 External links

[edit] 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.

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 other digits between the current digit and the decimal point. If the current digit is on the left hand side of the decimal point (ie, it is greater than 1) then n is positive; if the digit is on the right hand side of the decimal point (ie, it is less than 1 or 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:

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

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:

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

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
   00000    00000
   00000    00000           00000   00000
   00000    00000     +                        +         0
   00000    00000           00000   00000
   00000    00000

241 in base 8:
   2 groups of 8² (64)          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

[edit] 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:

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

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

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

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

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

Since a complete infinite string of digits can not 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:

$2.42\overline{314}_5 = 2.42314314314314314..._5$

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:

$0.1_3\,$

$0.\overline3_{10} = 0.3333333..._{10}$

$0.\overline{01}_2$

$0.2_6\,$

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 zero's can be appended:

$3.46_7 = 3.460_7 = 3.460000_7 = 3.46\overline0_7$

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):

$3.46_7 = 3.45\overline6_7$

$1_{10} = 0.\overline9_{10}$

$220_5 = 214.\overline4_5$

[edit] 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.

[edit] 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
   00000    00000
   00000    00000           00000   00000
   00000    00000     +                        +         0
   00000    00000           00000   00000
   00000    00000

is equal to 107 in base 8:
    1 group 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:

$\sum_{i=0}^n \left( a_i\times B^i \right)$

Thus, in the example above:

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

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.

[edit] 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.

[edit] 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.

[edit] 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).

[edit] 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 relates to one and only one octal digit.

[edit] See also

[edit] References

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

[edit] External links

MathWorld entry on base

Retrieved from "http://en.wikipedia.org../../../r/a/d/Radix.html"

Categories: Elementary mathematics | Numeration

Radix

From Wikipedia, the free encyclopedia

Contents

[edit] System

[edit] Infinite representations

[edit] Relationship between real numbers and their representations

[edit] Conversion among bases

[edit] Applications

[edit] Historical systems

[edit] Other bases in human language

[edit] Computing

[edit] See also

[edit] References

[edit] External links

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