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 number 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. In certain non-standard positional numeral systems, the definition of the base deviates from the above.

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.

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 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.

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.

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

4\times 10^2 + 6\times 10^1 + 5\times 10^0

Numbers that are not integers use places beyond a 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

Relationship between real numbers and their representations

It has been demonstrated that there is a one-to-one correspondence between real numbers and their representations in any base except the trivial 0 or 1. That is, given a real number x and a base b, we can find only one function (or vector) f(x, k) which retrieves the kth digit, where k = 0 is the units digit, k = ...,-1,0 are the fractional parts, and k = 1, ... are the whole registers (10s, 100s, etc. for b = 10). In other words, each real number has exactly one infinite decimal representation in any base. Also, each such representation converges to a real number. (The latter fact is very easily justified using the so-called completeness axiom).

Another, stronger result states that every rational number has a repeating fraction representation in any base b: that is, for each rational number we can find a representation in any legitimate base such that the sequence of digits repeats with a fixed period after some nth digit. Every such representation converges to a rational number.

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 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 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

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.

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.

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.