Positional notation

A place value system, position system or polyadic number system is a number system in which the (additive) valence of a symbol depends on its position, the place. For example, in the widespread decimal system for the example value "127" the digit "1" has the value 1 - 100, in addition for the digit "2" the value 2 - 10 as well as for the "7" 7 - 1 - the symbols "1", "2" and "7" have a valence which depends on which position/place they stand in the number. Assuming a finite supply of symbols (usually called digits or number signs, in the example "0"... "9"), the number of digits required depends logarithmically on the size of the number represented - unlike addition systems where this relationship is linear (asymptotically, i.e. for very large numbers).

The size of the digit set plays a crucial role. In the tens system example, the digit set is "0" to "9", which is b=10 different symbols. In the important integer systems, the value of the number represented is equal to the sum of the products of the respective digit value with its place value, i.e. a polynomial in values of the digits as coefficients; in the example 127:

Numerical value = "1" - ¹⁰² + "2" - ¹⁰¹ + "7" - ¹⁰⁰.

Therefore called the base or fundamental number of the system. The representation of numbers with respect to a base often called their -adic representation (not to be confused with -adic numbers). Any integer $b\geq 2$ suitable as a base for a place value system.

Examples of place value systems are the decimal system (decadic system with base 10) commonly used in everyday life, the dual system (dyadic system with base 2) frequently used in data processing, the octal system (with base 8), the hexadecimal system (with base 16), and the sexagesimal system (with base 60). An example of a number system that is not a place value system is that of Roman numerals. This is an addition system.

A binary clock can use light emitting diodes to represent binary values. In the picture above, each column of LEDs is a BCD coding of the traditional sexagesimal time representation.

History

Basic Terms

In a place value system, numbers are represented with the help of digits and, if necessary, signs or separators. The value of a number results from the arrangement of the digits from their digit values and place values.

Base

The total number of digits is called the base of the place value system. A place value system with base is also called -adic number system. The most common bases are:

: the dual system used in digital technology
: our familiar decimal system
: the hexadecimal system which is important in data processing.

For other used in practice. -adic number systems used in practice, see the section Common bases.

Digit set

In a digit system, a digit system with exactly different digits is used. In the most common digit systems, a digit of the type given below stands for an integer digit value ∈ $\in \{0,1,\ldots ,b-1\}$ . Counting up (which is equivalent to adding a one) proceeds to the digit with the next highest value in the specified order; however, with the few digits available, only a few counting steps would be possible. Therefore, for the most significant digit, a carryover is made to the least significant digit by adding a one, and a one is added to the next highest digit. In case of a carry-over to an unoccupied digit, this digit is filled with a zero in advance; in case of an unlimited number of digits, counting can thus be continued without limitation.

In common number systems, the following digits are used and assigned a digit value (for better distinction, digit symbols are printed here in bold and their associated values are printed normally):

In the dual system, the two digits 0 and 1 are used and the values of the numbers 0 and 1 are assigned to them respectively.
In the decimal system, the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used and the values of the numbers from 0 to 9 are assigned to each of them in the conventional order.
In the hexadecimal system, the sixteen digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F are used and the values of the decimal numbers from 0 to 15 are assigned to each of them.

If the base is very large, it usually comes to a combination of a few digits in another number system. Thus it is usual with the sexagesimal system to use a decimal number from 0 to 59 as "digit" instead of 60 different characters. IP addresses in the IPv4 format consist of 4 "digits" that can take values from 0 to 255 and are separated by a dot, for example 192.0.2.42. A different way of assigning digit to digit value was chosen for the Base64 encoding.

Sometimes other symbols are used instead of digits; for example, in electronics the two states of a dual system are often not described by 0 and 1, but h and l (for "high" and "low" voltage values) are used instead (rarely o and l for "on" and "low").

Place and value

The value of a number now results from the arrangement of the digits in a sequence of digits. Each place that a digit occupies or should occupy in this arrangement is a digit. Each digit is assigned a place value that corresponds to a power of the base. The digit with the lowest place value is placed on the far right. In the decimal system, for example, the following applies to the representation of natural numbers:

The place value of the first digit from the right ("ones digit") is $10^{0}=1$ .
The place value of the second digit from the right ("tens digit") is $10^{1}=10$ .
The place value of the third digit from the right ("hundreds digit") is $10^{2}=100$ , and so on.

It proves to be advantageous to number the digits not from one but from zero. In this way the -th digit has just the place value $b^{i}$ . In the representation of rational numbers, negative exponents are also allowed.

Representations of different types of numbers

Representation of natural numbers

Natural numbers are represented in the -adic representation by a finite sequence of digits in the form

$\mathbf {a} _{n}\ldots \mathbf {a} _{2}\mathbf {a} _{1}\mathbf {a} _{0}$

is displayed. This sequence of digits is now assigned the number

$Z=\sum _{i=0}^{n}a_{i}\cdot b^{i}=a_{0}\cdot b^{0}+a_{1}\cdot b^{1}+a_{2}\cdot b^{2}+\dotsb +a_{n}\cdot b^{n}$

where $a_{i}$ the digit value assigned to the digit $\mathbf {a} _{i}$ is the digit value assigned to it.

It can be shown that for every natural number there exists a sequence of digits whose associated value is In general, there are even several sequences. It is sufficient to prefix the digit 0 = 0 arbitrarily often on higher-order digits. If sequences with leading 0 are forbidden, it can be shown that this assignment is even one-to-one, that is, for every natural number there exists exactly one sequence whose assigned value is As an exception to this prohibition, the number 0 is not assigned the empty sequence (i.e. the sequence without a single member), but the sequence with exactly one digit, namely the one to which the value 0 is assigned (i.e. 0), in order to make this number typographically recognizable.

As an example of the specified number representation, let's consider the sequence of digits 694 in the decimal system ( b=10 ). It stands for:

$4\cdot 10^{0}+9\cdot 10^{1}+6\cdot 10^{2}=694.$

The digit sequence 2B6 in the hexadecimal system ( b=16 ) stands for $a_{0}+a_{1}\cdot b+a_{2}\cdot b^{2}$ with $a_{0}$ = 6 = 6; $a_{1}$ = B = 11; a_2 = 2 = 2.

So the sequence 2B6 has the value of the decimal number

$6+11\cdot 16+2\cdot 16^{2}=6+176+512=694.$

Accordingly, the sequence of digits 1010110110 in the dual system ( b=2 ) has the value of the decimal number

$0\cdot 2^{0}+1\cdot 2^{1}+1\cdot 2^{2}+0\cdot 2^{3}+1\cdot 2^{4}+1\cdot 2^{5}+0\cdot 2^{6}+1\cdot 2^{7}+0\cdot 2^{8}+1\cdot 2^{9}=2+4+16+32+128+512=694.$

Representation of integers

In a system consisting of a positive base and a purely non-negative set of digits, negative numbers cannot be represented. In such systems a minus sign ("-") is added, which is prefixed to the number constants if necessary. This is accompanied by a slight loss of uniqueness, since the number 0 can be written as a signed zero in the form +0, -0, or even ±0. Representations of numbers other than 0 that are not preceded by a minus sign are interpreted as positive numbers. Sometimes, however, one wants to emphasize this positivity (e.g., if the number is to be identified as an increment). In such cases, a plus sign ("+") is prefixed in the display.

Representation of rational numbers

The notation is extended into the negative exponents of the base by connecting the corresponding digits to the right of a separator added for this purpose in a gapless sequence. In German-speaking countries (except Switzerland), the comma "," is used for this purpose, whereas in English-speaking countries the period ". " is commonly used. The values of the digits after the separator are multiplied by $b^{-i}$ , where indicates the position after the comma. For example, the rational number 1+3/8 = 1.375 is represented in the 2-adic place value system by the digit sequence 1.011. In fact

$1\cdot 2^{0}+0\cdot 2^{-1}+1\cdot 2^{-2}+1\cdot 2^{-3}=1+0/2+1/4+1/8=1+3/8.$

After adding the separator, many rational numbers -adic, but by no means all, for it may happen that an infinite sequence of decimal places is required for the representation, which is then periodic. Usually this period is marked by a line drawn across the repeating digits, thus marking the length of the period and allowing a (finite) writeup without dots.

While the number 1/5 = 0.2 has the finite digit sequence 0.2 in the decimal system, its representation in the dual system is periodic:

0,00110011…₂ = 0,00112.

In contrast, the digit sequence 0.1 in the 3-adic (ternary) system means the rational number 1-3-1 = 1/3, which in the decimal system corresponds to an infinite periodic digit sequence 0.333... = 0.3dec.

Provided that 0 is a digit and that for every integer there is a digit whose value ${\text{mod }}b$ is congruent to it (which is always the case for standard digit systems), it is generally true that a fraction has a finite -adic representation if, after truncation, all prime factors of its denominator are also prime factors of $b=:p_{1}^{\nu _{1}}\cdots p_{k}^{\nu _{k}}$ (for $p_{1},\ldots ,p_{k}\in \mathbb {P}$ and ν $\nu _{1},\ldots ,\nu _{k}\in \mathbb {N}$ ) are. (Thus, for a finite representation in the decimal system, the truncated denominator must be a product of the numbers two and five. Exactly then the fraction is a decimal fraction in the strict sense or becomes one by extending).

The finite representations form the ring

$\mathbb {Z} _{S}:=\{x\in \mathbb {Q} \mid \exists i\in \mathbb {N} _{0}:xb^{i}\in \mathbb {Z} \}=\mathbb {Z} b^{\mathbb {Z} }$ ,

where stands $S:=\{p_{1},\ldots ,p_{k}\}\subset \mathbb {P}$ for the set of prime factors of For these rational numbers, in a fully truncated fraction representation, the denominator has only prime divisors $p_{i}\in S$ . For any nonempty the subring is $\mathbb {Z} _{S}$ of $\mathbb {Q}$ (like $\mathbb {Q}$ itself) dense in both $\mathbb {Q}$ as well as in $\mathbb {R}$ , i.e., any real number can be approximated arbitrarily exactly by numbers from $\mathbb {Z} _{S}$ .

If one considers only representations of finite length, then already the digit sequences 1, 1,0, 1,000 in the decimal system all designate the same rational number 1 (not to mention the representations 01, 0001 with leading zeros). These ambiguities can still be suppressed by prohibitions of leading and trailing zeros. If, however, the infinite representations belong to the system from the beginning, then the non-terminating representation 1.000... = 1.0 and, in addition, the completely different looking representation 0.999... = 0.9 (all with the value 1) are added, see the article 0.999.....

Normally, misunderstandings are not to be feared, so that one can allow both representations. However, uniqueness is required, for example, for the Z-curve, which is $Z\,\colon \mathbb {R} ^{2}\to \mathbb {R} ^{1}$ maps injectively and where two -digit sequences are alternately squeezed into one. Incidentally, the discontinuity points of the function are precisely the arguments that have a finite -adic representation.

The -adic representation of a truncated fraction $q=:z/(m\cdot n)$ with $z\in \mathbb {Z} ,m=p_{1}^{\mu _{1}}\cdots p_{k}^{\mu _{k}}$ and $n\in \mathbb {N}$ divisible to the base has period length 0 for n=1 , so it is finite. Otherwise, an element of the prime residue class $[b]\in \mathbb {Z} _{n}^{*}$ such that $b^{\varphi (n)}\equiv 1{\text{ mod }}n$ (with φ $\varphi$ as the Eulerian φ-function). The -adic period length of the truncated fraction is then the smallest exponent $\operatorname {ord} _{n}(b):=e>0$ , for which is a divisor of $b^{e}-1$ (See also the section Algorithm for rational numbers and the article Rational number#decimal fraction development).

Representation of real numbers

The representation of real numbers takes place in principle exactly the same as that of rational numbers by b-adic development. With rational numbers this supplies a terminating or an infinite periodic digit sequence.

The b-adic expansion of an irrational number (like π or ${\sqrt {2}}$ ), on the other hand, always yields an infinite non-periodic sequence of digits. By lengthening the decimal part, an arbitrarily exact approximation to the irrational number is possible.

As in the case of rational numbers with an infinitely periodic sequence of digits, a finite representation for irrational numbers is possible by introducing new symbols, as has been done here for the examples π and ${\sqrt {2}}$

Nevertheless, even with an arbitrary but finite number of additional characters, not every real number can be represented as a finite string. This is because the set of real numbers is overcountable, but the set of all finite representations with finite character set is only countable.

But if the "representation" of a real number is understood as the sequence of digits resulting from the b-adic development, then every real number is representable as a (possibly infinite) b-adic fraction, even if not every such fraction is actually recordable.

Formulas

Calculation of a digit value

The last digit of the -adic representation of a natural number is the remainder of when divided by . This remainder is also represented by the expression

$n-b\left\lfloor {\frac {n}{b}}\right\rfloor$

given; where ⌊ denotes $\lfloor {\cdot }\rfloor$ the Gaussian bracket. More generally, the bracket formed by the last digits of the remainder of when divided by $b^{k}$ .

The digit z_k at the -th digit (counting from the right at the ones digit starting with zero and progressing to the left) of a positive real number is

$z_{k}=\left\lfloor {\frac {x}{b^{k}}}\right\rfloor -b\left\lfloor {\frac {x}{b^{k+1}}}\right\rfloor .$

Here $z_{k}\in \{0,1,\dotso ,b-1\}$ is an element of the standard digit set. If we add negative for which the corresponding (negative) decimal place results, then one has

$x=\sum _{k=-n}^{\infty }z_{-k}b^{-k}$

with sufficiently large $n:=\lfloor \log _{b}(x)\rfloor .$

Algorithm for rational numbers

For rational $0<x=p/q<1$ (and a base $b\in \mathbb {N} _{>1}$ the above formula can be embedded in the following algorithm:

function b_adic(b,p,q) // b ≥ 2; 0 < p < q static digit string = "0123..."; // up to the character with value b-1 begin s = ""; // the string to be formed pos = 0; // here are all digits to the right of the decimal point while not defined(occurs[p]) do occurs[p] = pos; // the number of the digit with remainder p bp = b*p; z = floor(bp/q); // index z of the digit in the supply: 0 ≤ z ≤ b-1 p = bp - z*q; // p integer: 0 ≤ p < q if p = 0 then pl = 0; return (s); end if s = s. substring(digit set, z, 1); // append digit from the digit set. // substring(s, 0, 1) is the first digit after the decimal point pos += 1; end while pl = pos - occurs[p]; // the period length (0 < pl < q) // mark the digits of the period with an overline: for i from occurs[p] to pos-1 do substring(s, i, 1) = overline(substring(s, i, 1)); end for return (s); end function

The first line highlighted in yellow corresponds to the digit calculation of the previous section.

The following line calculates the new remainder $p'$ of the division modulo the denominator . The Gaussian bracket floor has the effect that

$bp/q-1\;\;<\;\;z=\lfloor bp/q\rfloor \;\;\leq \;\;bp/q.$

It follows that $bp-q<zq\;\implies \;p':=bp-zq<q$ z qtaken together 0 ≤ $zq\leq bp\;\implies \;0\leq bp-zq=:p',$ $0\leq p'<q.$ Thus, since all remainders are integer non-negative and smaller than so there are only many different of them, they must recur in the while loop. The recurrence of a remainder determined by the existence of the associative data field occurs[p].

The period of the digits has the same length as the period of the remainders. (For more details on the period length see above).

Calculation of the number of digits

The number of digits of the -adic representation of a natural number $n \in \N_0$ is

$a={\begin{cases}1,&{\text{wenn }}n=0,\\\lfloor \log _{b}{n}\rfloor +1,&{\text{wenn }}n\geq 1.\end{cases}}$

Add a digit

If you append to the -adic representation of a number the far right, we obtain the -adic representation of the number $n\cdot b+z$ .
If, on the other hand, the number prefixed on the far left with obtain the -adic representation of the number $z\cdot b^{a}+n$ , where is the number of digits of as given above.

Common bases

The best known and most widespread place value system is the decimal system (tens system) with base 10 and digits 0 to 9. The decimal system originated in India. The Persian mathematician Muhammad ibn Musa al-Chwarizmi used it in his arithmetic book, which he wrote in the 8th century. As early as the 10th century, the system was introduced in Europe, at that time still without zero. However, it did not become established until the 12th century with the translation of the aforementioned arithmetic book into Latin. The BCD code is used to store decimal digits in the computer.
In the 17th century, the mathematician Gottfried Wilhelm Leibniz introduced the dual system (binary number system) with dyadics, i.e. the place value system with the base 2 and the digits 0 and 1. This is used primarily in information technology, since its logic is based solely on bits, which are either true or false or 1 or 0.
Since binary representations of large numbers are confusingly long, the hexadecimal or sedecimal system is often used in their place, which works with the base 16 (and the digits 0, 1, ..., 9, A, B, ..., F). Hexadecimal and binary representation can be easily converted into each other, because one digit of a hexadecimal number corresponds exactly to 4 digits (= 1 nibble) of a binary number.
In computer technology, in addition to the binary and hexadecimal systems, the octal system with base 8 (digits 0 to 7, three binary digits = one octal digit) is also used. However, this use is decreasing more and more because the word lengths of eight bits common today cannot be converted into a whole number of digits in the octal system.
Also used is base 64 at Base64 (with unusual symbol order); base 62 at Base62 with the digits 0 to 9, A to Z, and a to z; and occasionally base 32 with the digits 0 to 9 and a to v under the designation Radix32.
From approx. 1100 B.C. in the Indo-Chinese area abacus (arithmetic table) were used, which are based on a unary system. But see above for the unary system in blocks of five, which, however, is an addition system.
The vigesimal system uses 20 as a base. It probably originated because the toes were used for counting in addition to the fingers, and was common in almost all Mesoamerican cultures, among others. The most developed system of this kind was used by the Maya in the Classical Period for astronomical calculations as well as for the representation of calendar dates. It was a place value system "with a leap", because in the second place only the digits from 1 to 18 occur, thus reaching 360 (approximate length of the solar year) as the third place value. The Maya knew the zero and used it also in their calendars.
The Indians of South America used number systems in base 4, 8 or 16, because they calculated with hands and feet, but did not include the thumbs.
The duodecimal system has 12 as its basis. We find it in the calculation with dozen and gros and in the Anglo-Saxon system of measurement (1 shilling = 12 pence) (see also Old weights and measures). The counting of hours also has its origin in this system. In many polytheistic religions there were 12 principal gods, divided, for example, in ancient Egypt into three supreme gods and 3 × 3 assigned gods. (Three was considered a perfect number; see also Trinity).
The Babylonians used a number system with base 60 (sexagesimal system; see also History of Weights and Measures).
A possibly expected number system to the base five with peoples, who use only one hand for counting, was not discovered so far. In Bantu languages, however, the names of the numbers 6, 7, 8 and 9 are often foreign words or can be understood as 5 + 1, 5 + 2, 5 + 3, 5 + 4, which points to a number system in base 5.

For example:
Swahili: 1 = moja, 2 = mbili, 3 = tatu, 4 = nne, 5 = tano, 6 = sita, 7 = saba, 8 = nane, 9 = kenda (Arabic: 6 = sitta, 7 = saba'a)
Tshicheva: 1 = modzi, 2 = wiri, 3 = tatu, 4 = nai, 5 = sanu, 6 = sanu ndi-modzi, 7 = sanu ndi-wiri, 8 = sanu ndi-tatu, 9 = sanu ndi-nai.

The quinary system is particularly pronounced in the South American Betoya: 1 = tey, 2 = cayapa, 3 = tozumba, 4 = cajezea, 5 = teente, 10 = caya ente, 15 = tozumba-ente, 20 = caesea ente.

The senary system is suitable for counting to thirty-five with 2 × 5 fingers. Linguistic traces of such a system are very rare (for example Breton 18 = triouec'h, about "3 6s").
The earlier assumption that the Maori use a base 11 system is now considered outdated. Some peoples use the base 18 system.

Conversions

Sometimes one needs conversions between place value systems. If the decimal system is not involved, one can use it as an intermediate step. The following calculations can also be done with the help of a pocket calculator, where usually the number input and output is done in the decimal system only.

Especially when numbers are to be converted from one system to another, it is common and convenient to identify the digit sequences by a subscript suffix $_{b}$ the base the number system used. Here, a missing suffix and the suffix ₁₀ stand by default for the conventional decimal representation, explicitly also _dez or _dec. The suffixes ₂ or _b denote binary and ₁₆ or _h hexadecimal represented numbers. Furthermore, the default set $\{0,1,\ldots ,b-1\}$ assumed to be the digit set. Occasionally, the labeled digit set is enclosed in square brackets.

There are two main variants

the iterated Euclidean division starting at the low significance digits, and
the evaluation of the digit polynomial e.g. in a kind of the Horner scheme. The smallest number of multiplications is needed if one starts at the most significant digit.

The selection is best based on which procedure is easiest to perform on the existing calculator.

Example 1: Converting a base 10 representation to a base 12 representation

A number has the decimal representation 4711. The search is for its representation in the system of twelve.

To obtain this representation, divide the given representation stepwise by the new base 12. The remaining remainders provide the representation in base 12. The first remainder corresponds to the lowest digit value of the new representation we are looking for (in our case, the digit $12^{0}$ ), the second remainder corresponds to the second lowest digit value (i.e., the digit $12^{1}$ ), and so on. The corresponding calculation is therefore:

4711 divided by 12 gives 392 remainder 7 (corresponds to the digit to the position $12^{0}$ in the result)
0392 divided by 12 gives 032 remainder 8 (corresponds to the digit to the position $12^{1}$ in the result)
0032 divided by 12 gives 002 remainder 8 (corresponds to the digit to the position $12^{2}$ in the result)
0002 divided by 12 gives 000 remainder 2 (corresponds to the digit to the position $12^{3}$ in the result)

As a duodecimal representation of the given number we thus obtain 2887. The conversion to other place value systems is done analogously.

Example 2: Converting a base 16 representation to a base 10 representation

Regarding the hexadecimal system with the digits 0, 1, ..., 9, A (value 10), B (value 11), C (value 12), D (value 13), E (value 14) and F (value 15) a number has the representation AFFE. We are looking for the representation of this number in the decimal system.

To obtain this representation, multiply the digit values of the given representation by the respective digit values and add up the results. The corresponding calculation is therefore:

10 (A) times $16^{3}$ gives 40960
15 (F) times $16^{2}$ gives 3840
15 (F) times $16^{1}$ gives 240
14 (E) times $16^{0}$ gives 14

As a decimal representation of the given number, we thus obtain $40960+3840+240+14=45054$ . The conversion to other place value systems is done analogously.

Example 3: Decimal places

With respect to the decimal system, a number has the representation 0,1. What we are looking for is the representation of this number in the dual system.

For this purpose, the fraction after the decimal point is multiplied repeatedly by the base of the target system. If a value greater than 1 occurs, its integer part is added to the series of decimal places, otherwise a 0 is added to the decimal places. If an integer occurs as a multiplication result, the decimal fraction is completely determined, but often a period will occur.

The corresponding calculation is therefore:

0.1 times 2 gives 0.2 , so the first decimal place is 0
0.2 times 2 equals 0.4 , so the second decimal place is 0
0.4 times 2 equals 0.8 , so the third decimal place is 0
0.8 times 2 gives 1.6 , so the fourth decimal place is 1
0.6 times 2 gives 1.2 , so the fifth decimal place is 1
0.2 times 2 (no longer needs to be executed because a period has occurred)

As a result we get 0.0001100110011...

Balanced place value systems

Special place value systems are the balanced ones. They always have an odd base $b\in \mathbb {N}$ and use both natural and negative digit values, namely those from the set $\{-{\tfrac {b-1}{2}},\dotsc ,-1,0,1,\dotsc ,{\tfrac {b-1}{2}}\}$ . Often, the negative digits are indicated by an underscore. For example, in the balanced ternary system, a number is represented by the digits 1, 0, and 1, to which the values -1, 0, and 1 are assigned.

A balanced place value system has the following properties:

The negative of a number is obtained by exchanging each digit with its inverse counterpart.
The first digit other than 0 indicates the sign. The system therefore manages without a separate sign.
Rounding to the nearest integer is done by simply truncating at the decimal point.

The representation of the integers is unambiguous.

But there are rational numbers which cannot be represented uniquely. Let $\mathbf {t} :={\tfrac {b-1}{2}}$ be the largest digit and ${\underline {\!\!\mathbf {t} \!\!}}:=-\mathbf {t}$ the smallest, then for example.

$0{,}{\overline {\mathbf {t} }}\;=\;1{,}{\overline {\underline {\!\!\mathbf {t} \!\!}}}\;=\;{\frac {1}{2}}.$

Lexicographic order

With positive base $b\in \mathbb {N} \!\setminus \!\!\{\!1\!\}$ the usual order relation of real numbers is closely related to the lexicographic order of representing these numbers. -adic strings. More precisely:

There is an order homomorphism (an order-preserving mapping) ω $\omega$ , which maps the arbitrarily (even infinitely) long strings in a -adic way into a real interval.
For no -adic system, ω is $\omega$ injective.
Which real numbers have multiple representations (multiple primal images) depends on the digit values of the associated digit system $\Sigma$ Its set is a subset of the rational numbers, so it has countable power. It lies densely in the image interval.

Derivation

Let $2\leq b\in \mathbb {N}$ and $\Sigma :=\{z_{1},z_{2},\ldots ,z_{b}\}$ be a strictly totally ordered alphabet with $z_{k}\prec z_{k+1},$ whose order relation is $\prec$ denoted by Furthermore, let $S,T\in \Sigma$ two characters with $S\prec T$ , then lexicographically

$Ss\prec Tt$

for all strings $s,t\in \Sigma ^{\infty }$ with $\Sigma ^{\infty }$ as the set of arbitrarily (even infinitely) long strings over $\Sigma$ (including the Kleene hull $\Sigma ^{*}$ of $\Sigma$ ).

The strings can also be understood as -adic representation, namely the values

$w\,\colon \Sigma \to \mathbb {Z}$

of the digits defined without gaps, i.e. $z_{k}$

$w(z_{k+1})=w(z_{k})+1$ ,

such that $\Sigma$ a minimal digit set for a -adic system and $w(z_{b})=w(z_{1})+b-1$ . We restrict ourselves to digit values whose magnitude is not greater than the base, so $w\,\colon \Sigma \to [-b,+b]$ (which covers the most important cases occurring in practice). The digits $S,T\in \Sigma$ can be chosen such that $w(S)=w(T)-1$ . This is compatible with $S\prec T$ , and the above lexicographic inequality remains valid even if the chains $s:=z_{b}z_{b}z_{b}\ldots$ and $t:=z_{1}z_{1}z_{1}\ldots$ have periods continued to infinity.

For the evaluation of the strings according to the -adic system one needs a continuation

$\omega \,\colon \Sigma ^{\infty }\to [-1,1]$

of the value function with ω $\omega (z)=w(z)/b\;\in [-1,1]$ for $z\in \Sigma$ and with

$\omega (x_{1}x_{2}x_{3}\ldots )=\sum _{i=1}^{\infty }w(x_{i})b^{-i}$ .

With respect to the metric of the ordinary Archimedean absolute value, the series converge to

$\omega (s)=\omega (z_{b}z_{b}z_{b}\ldots )=\sum _{i=1}^{\infty }w(z_{b})b^{-i}={\frac {w(z_{b})}{b-1}}$

and

$\omega (t)=\omega (z_{1}z_{1}z_{1}\ldots )=\sum _{i=1}^{\infty }w(z_{1})b^{-i}={\frac {w(z_{1})}{b-1}}$ ,

and it is

$\omega (s)={\frac {w(z_{b})}{b-1}}={\frac {w(z_{1})+b-1}{b-1}}={\frac {w(z_{1})}{b-1}}+1=\omega (t)+1$ .

Thus, although lexicographically

$Ss\prec Tt$

(and the strings are obviously different in $\Sigma ^{\infty }$ ), but they are reduced to the same real number

$\omega (Ss)=(w(S)+\omega (s))/b=(w(T)-1+\omega (t)+1)/b=(w(T)+\omega (t))/b=\omega (Tt)\;\in [-1,1]$

is mapped. Thus ω is $\omega$ not injective.

If one includes equality in the order relations, then the following applies

$Ss\preceq Tt\quad \implies \quad \omega (Ss)\leq \omega (Tt)$

and ω $\omega$ is an order homomorphism, but it is not bijective and thus not an order isomorphism.

In the section Representation of rational numbers $\mathbb {Z} b^{\mathbb {Z} }$ was worked out to be the set of real numbers with finite representation. The set of real numbers with multiple representation is then

$\mathbb {Z} b^{\mathbb {Z} }+{\frac {w(z_{1})}{b-1}}$ ,

thus for $w(z_{1})=0$ same as that of finite representations; so for many common -adic systems.

Generalizations

Number systems with mixed bases

An obvious generalization is to choose different bases for the different digit positions. We then speak of number systems with mixed bases. A few interesting examples are:

alternating a or b, where a and b are two different natural numbers > 1
2 or 3 but in order, so that $e^{k}$ is approximated "relatively closest" with the product of the first k bases
the natural numbers > 1 in sequence are used as basis ("faculty basis")

the prime numbers in sequence or the (then repeating) prime numbers that occur with each next higher prime number power (see also representation of profinite numbers with multiple bases)

In the last two cases, you basically have an infinite number of different digit symbols to provide.

Date format as number system with mixed bases

→ Main article: Date format

Also the representation of date and time has traditionally several bases and number systems. In this context, the only example is the following representation commonly used in the Anglo-Saxon language area

[1-12] [1-31] [0-9]^{[2,4,*] [}1-12] [am,pm] [0-59] [0-59] [0-9]^*

where, in addition, the order of year, month and day on the one hand and half day and hour on the other hand are reversed against the order of precedence. So here the bases 2, 10, 12, 28-31 and 60 are used. In particular, it is remarkable that the base of the day digit is based on the value of the month digit.

Non-natural numbers as basis

The base not necessarily have to be a natural number. All (also complex) numbers with magnitude greater than 1 can be used as base of a place value system.

Negative bases

Digit systems with negative bases $b\in \mathbb {Z}$ with $b\leq -2$ cooperate with the same digit sets as their positive counterparts $-b\in \mathbb {N}$ and is often called a radix. They are often denoted by the prefix nega-, e.g., the negadecimal, negabinary, negaternary, etc. place value system.

These place value systems do not require an extra sign. On the other hand, the representations often require one digit more than in the corresponding system with positive base. Furthermore, the arithmetic operations, especially the arithmetic comparison and the formation of the absolute value, are somewhat more complex.

If the set of digits is minimal, e.g. $\{0,1,\ldots ,-b\!-\!\!1\}$ , then all integers are uniquely representable. As with positive bases, there are rational numbers that are not uniquely representable. Let

$T:=0{,}{\overline {01}}_{b}=\sum _{i=1}^{\infty }b^{-2i}={\frac {1}{b^{2}-1}}$

and $\mathbf {t} :=-b\!-\!\!1$ is the largest digit, then both

$0{,}{\overline {0\mathbf {t} }}_{b}=\mathbf {t} T={\frac {-b\!-\!\!1}{b^{2}-1}}={\frac {1}{-b+1}}$

as well as

$1{,}{\overline {\mathbf {t} 0}}_{b}=1+\mathbf {t} bT={\frac {(b^{2}-1)+(-b\!-\!\!1)b}{b^{2}-1}}={\frac {1}{-b+1}}.$

Some arithmetic operations are brought by the English-language article.

Irrational bases

If one wants to represent all real numbers, then for non-integer or irrational basis $b\in \mathbb {R} \setminus \mathbb {Z}$ the minimum size of the numeral system ⌈ $\lceil |b|\rceil$ (magnitude dashes and Gaussian brackets). For such generalized place value systems, some of the statements made here about the finite representability of rational numbers do not apply.

For example, if the golden ratio $\Phi ={\frac {1+{\sqrt {5}}}{2}}$ used as the base and $\left\{0,1\right\}$ as the digit set, s,t represents an integer or an irrational number of the form $s+t\cdot {\sqrt {5}}$ with rational Nevertheless, not every such number has a finite representation.

A representation also based on the golden ratio is the Zeckendorf representation, in which, however, not the powers of $\Phi$ but the Fibonacci numbers are taken as place values.

Non-real bases

The first number system to represent a complex number as a single sequence of digits rather than as two separate sequences of digits - one each for real and imaginary parts - was the "quater-imaginary" system proposed by D. Knuth in 1955. It has $2\mathrm {i}$ as base and 0, 1, 2, 3 as digits. There, for example, $-1=103_{2\mathrm {i} }$ and $\mathrm {i} =10{,}2_{2\mathrm {i} }$ . See also the English-language article en:Quater-imaginary base.

Another system was proposed in 1964 by S. Khmelnik and worked out for digital machinery. It has $\mathrm {i} -1$ as base and 0, 1 as digits. For example, $-1=11101_{\mathrm {i} -1}$ and $\mathrm {i} =11_{\mathrm {i} -1}$ . See also the English-language article en:Complex base systems.

p-adic numbers

→ Main article: p-adic number

The place value systems presented here are based on convergence with respect to the metric of the ordinary Archimedean absolute value. The infinite series - which here always converge, namely "to the right" at the small powers of the base (exponents $\searrow -\!\infty$ ) - are then real (or complex) numbers. However, there are metrics for the rational numbers that are based on nonarchimedean magnitude functions and allow a very similar notation with base and digit set. The infinite series - which always $\nearrow +\!\infty$ converge there, too, according to the convention "left" at the large powers (exponents - are p-adic numbers.

Although finite -adic expressions agree with the same sequence of digits in (then also finite) p=:b -adic representation, but there are serious differences from the (Archimedean) systems otherwise presented here. The most important ones are:

The -adic representations are always (reversibly) unique.
A sign is not required. The representation of as an infinite sum is ${\textstyle -1=\sum _{i=0}^{\infty }(p-1)\cdot p^{i}}$ .
A -adic ring cannot be ordered.
If is decomposable, that is, not a prime number, then the -adic ring contains $\mathbb {Z} _{p}$ Zero divisors (all of which have non-terminating representations). Details in Profinite number#10-adic numbers.
The non-terminating series represent number objects with completely different arithmetic properties in both systems. The periodic ones among them represent rational numbers in both systems.
All algorithms for the basic arithmetic start on the right at the small exponents (possibly negative, but $>-\!\infty$ , like the powers and carries, run in the same direction to the left to the large exponents. If the calculation is aborted, the size of the error can be specified immediately.

Related texts

The article Divisibility explains how in the representation of place value systems in certain cases it can be recognized whether a number is a divisor of another. Cantor's normal form generalizes the representation of numbers in the place value system to ordinal numbers.

An example of application shows the Berlin clock.

Questions and Answers

Q: What is place value in mathematics?

A: Place value is the value represented by a digit in a number.

Q: What is the decimal system?

A: The decimal system is a numerical system that is based on ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Q: What are the columns to the left of the decimal place in the decimal system?

A: The columns to the left of the decimal place in the decimal system are the units column, the tens column, and the hundreds column.

Q: What are the columns to the right of the decimal place in the decimal system?

A: The columns to the right of the decimal place in the decimal system are the tenths column, the hundredths column, and the thousandths column.

Q: Can you give an example of the place value of the digit 5 in the number 5,382?

A: The place value of the digit 5 in the number 5,382 is in the hundreds column.

Q: Which column represents the thousands place in the decimal system?

A: The thousands place is to the left of the hundreds place.

Q: Can you illustrate how the columns in the decimal system are arranged?

A: In the decimal system, the columns to the left of the decimal place are arranged in the order of hundreds, tens, and units, while the columns to the right of the decimal place are arranged in the order of tenths, hundredths, and thousandths.