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Encyclopedia > Base (mathematics)

In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... A numeral is a symbol or group of symbols that represents a number. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ...


The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system (except in bijective numeration). The various positional numeral systems differ from one another only in the radix they use. The base itself is almost always expressed in decimal notation. Bijective numeration is any numeral system that establishes a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits. ...


Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, 238 indicates that the number 23 is expressed in base 8 (and is equivalent in value to the decimal number 19). This notation will be used in this article. A subscript is a number, figure or indicator, that appears below the normal line of type, when used in a formula, mathematical expression or description of a chemical compound. ...

Contents


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. Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to... The octal numeral system is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ...


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. In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...


For example, the number 465 in its respective base (which is clearly at least base 7) is equal to:

4times 10^2 + 6times 10^1 + 5times 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. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... The decimal separator is used to mark the boundary between the integer and the fractional parts of a decimal numeral. ...


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 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 

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 anan-1...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_{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 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_{10} = 1times 8^2 + 0times 8^1 + 7times 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. The decimal (base ten or occasionally denary) numeral system has ten as its base. ...


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. Babylonia was an ancient state in Iraq), combining the territories of Sumer and Akkad. ...


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. Originally, the word computing was synonymous with counting and calculating, and a science that deals with the original sense of computing mathematical calculations. ... Look up binary in Wiktionary, the free dictionary. ... In mathematics and computer science, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ...


See also

A numeral is a symbol or group of symbols that represents a number. ... Mixed radix numeral systems are more general than the usual ones in that the numerical base may vary from position to position. ... In mathematics, radix point refers to the symbol used in numerical representations to separate the integral part of the number (to the left of the radix) from its fractional part (to the right of the radix). ... A Radix sort is a fast stable sorting algorithm which can be used to sort items that are identified by unique keys. ...

References

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

External links

  • MathWorld entry on base

  Results from FactBites:
 
base (mathematics) - Hutchinson encyclopedia article about base (mathematics) (317 words)
For example, the logarithm of 100 to the base ten is 2.
For bases beyond 10, the denary numbers 10, 11, 12, and so on must be replaced by a single digit.
Thus in base 16, all numbers up to 15 must be represented by single-digit ‘numbers’, since 10 in hexadecimal would mean 16 in decimal.
Base (mathematics) - Wikipedia, the free encyclopedia (1004 words)
In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system.
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.
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.
  More results at FactBites »

 
 

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