# Decimal system

A decimal system (from Middle Latin decimalis , to Latin decem “ten”), also known as the decimal system or decade system , is a number system that uses the number 10 as the base .

As a rule, it is specifically understood to mean the decimal system with place value system , which was developed in the Indian numerals , passed on to European countries through Arabic mediation and is now established worldwide as an international standard.

However, decimal systems are also number systems based on 10 without a place value system, as they are based on the numerals of many natural languages ​​and older numerals , sometimes in conjunction with quinary , vigesimal or other number systems .

The emergence of decimal systems as well as quinary systems is anthropologically associated with the number of fingers as a counting and arithmetic aid , an explanation that is also supported by the fact that in some languages ​​as numerals for 5 and 10 expressions with the meaning “hand "(5) or" two hands "(10) are needed.

## Decimal place value system

### Digits

Ten digits are used in the decimal system

0 (zero) , 1 (one) , 2 (two) , 3 (three) , 4 (four) , 5 (five) , 6 (six) , 7 (seven) , 8 (eight) , 9 (nine) ,

which are called decimal digits .

However, these digits are spelled differently in different parts of the world.

Indian numerals are still used today in the various Indian scripts ( Devanagari , Bengali script , Tamil script , etc.). They are very different from each other.

### definition

In the German-speaking world, a decimal number is usually in the form

${\ displaystyle z_ {m} \, z_ {m-1} \, \ ldots \, z_ {0} \, \ operatorname {,} \, z _ {- 1} \, z _ {- 2} \, \ ldots \, z _ {- n} \ qquad \ left (m, n \ in \ mathbb {N} \ quad z_ {i} \ in \ {0, \ ldots, 9 \} \ right)}$

written down; In addition, there are other spellings depending on the purpose and location . Each is one of the above-mentioned numbers. Each digit has a numerical value and, depending on its position, a place value. The digit value is in the conventional counting order. The index determines the value, which is the power of ten . The digits are written one after the other without separators, with the most significant digit with the digit on the far left and the lower significant digits with the digits up to the right of it in descending order. To represent rational numbers with non-periodic development, the digits to follow after a separating comma . In the English-speaking world, a point is used instead of the comma. ${\ displaystyle z_ {i}}$${\ displaystyle i}$${\ displaystyle 10 ^ {i}}$${\ displaystyle z_ {m}}$${\ displaystyle z_ {m-1}}$${\ displaystyle z_ {0}}$${\ displaystyle z _ {- 1}}$${\ displaystyle z _ {- n}}$

The value of the decimal number is thus obtained by adding up these digits, which are multiplied by their place value beforehand; In addition, the sign must be put in front; a missing sign means a plus: ${\ displaystyle Z}$

${\ displaystyle Z = \ sum _ {i = -n} ^ {m} z_ {i} \ cdot 10 ^ {i} \ geq 0 \; \ qquad Z = - \ sum _ {i = -n} ^ { m} z_ {i} \ cdot 10 ^ {i} \ leq 0}$.

This representation is also called decimal fraction development .

Example:

${\ displaystyle 7023 {,} 48 = 7 \ cdot 10 ^ {3} +0 \ cdot 10 ^ {2} +2 \ cdot 10 ^ {1} +3 \ cdot 10 ^ {0} +4 \ cdot 10 ^ {-1} +8 \ cdot 10 ^ {- 2}}$

With resolved potencies we get:

${\ displaystyle 7023 {,} 48 = 7 \ cdot 1000 + 0 \ cdot 100 + 2 \ cdot 10 + 3 \ cdot 1 + 4 \ cdot 0 {,} 1 + 8 \ cdot 0 {,} 01}$

### Decimal fraction development (converting periodic decimal numbers into fractions)

With the help of the expansion of the decimal fraction , you can assign a sequence of digits to every real number . Each finite part of this sequence defines a decimal fraction that is an approximation of the real number. The real number itself is obtained by moving from the finite sums of the parts to the infinite series over all digits.

Formally, the value of the series is called with. ${\ displaystyle z = 0, a_ {1} a_ {2} a_ {3} \ ldots}$${\ displaystyle \ textstyle \ sum _ {i = 1} ^ {\ infty} a_ {i} 10 ^ {- i}}$

It is said that the decimal fraction development stops when the sequence of digits from a position n consists of only zeros, i.e. the displayed real number is already a decimal fraction itself. Especially with all irrational numbers , the sequence of digits does not break off; there is an infinite decimal fraction expansion.

The following relationships are used to transform periodic decimal fraction expansion :

${\ displaystyle 0 {,} {\ overline {1}} = {\ frac {1} {9}}; \ quad 0 {,} {\ overline {01}} = {\ frac {1} {99}} ; \ quad 0 {,} {\ overline {001}} = {\ frac {1} {999}}; \ quad \ ldots}$.

These identities are apparent from the calculation rules for geometric series , which for applicable. In the first example one chooses and does not start the summation until the first term in the sequence. ${\ displaystyle \ textstyle \ sum _ {i = 0} ^ {\ infty} q ^ {i} = {\ frac {1} {1-q}}}$${\ displaystyle q <1}$${\ displaystyle q = 10 ^ {- 1}}$

Examples:

${\ displaystyle 0 {,} 55555 \ ldots = 0 {,} {\ overline {5}} = {\ frac {5} {9}}}$
${\ displaystyle 0 {,} 33333 \ ldots = 0 {,} {\ overline {3}} = {\ frac {3} {9}} = {\ frac {1} {3}}}$
${\ displaystyle 0 {,} 424242 \ ldots = 0 {,} {\ overline {42}} = {\ frac {42} {99}} = {\ frac {14} {33}}}$
${\ displaystyle 0 {,} 081081081 \ ldots = 0 {,} {\ overline {081}} = {\ frac {81} {999}} = {\ frac {3} {37}}}$

The period is transferred to the counter . There are as many nines in the denominator as there are digits in the period. If necessary, the resulting break should be shortened .

The calculation is a little more complicated if the period does not immediately follow the decimal point:

Examples:

${\ displaystyle 0 {,} 83333 \ ldots = 0 {,} 8 {\ overline {3}} = 8 {,} {\ overline {3}}: 10 = \ left (8 + {\ frac {3} { 9}} \ right): 10 = \ left (8 + {\ frac {1} {3}} \ right): 10 = {\ frac {25} {3}}: 10 = {\ frac {25} { 30}} = {\ frac {5} {6}}}$
${\ displaystyle 0 {,} 492363636 \ ldots = 0 {,} 492 {\ overline {36}} = 492 {,} {\ overline {36}}: 1000 = 492 {\ frac {36} {99}}: 1000 = 492 {\ frac {4} {11}}: 1000 = {\ frac {5416} {11}}: 1000 = {\ frac {5416} {11000}} = {\ frac {677} {1375}} }$
${\ displaystyle 13 {,} 54323232 \ ldots = 13 {,} 54 {\ overline {32}}}$
1st step: you multiply the starting number by a power of ten so that exactly one period (in the example 32) is in front of the decimal point:
${\ displaystyle x = 13 {,} 543232 \ ldots | \ cdot 10 {.} 000}$
${\ displaystyle 10 {.} 000 \ cdot x = 135 {.} 432 {,} 3232 \ ldots}$
2nd step: then you multiply the starting number by a power of ten so that the periods start exactly after the decimal point:
${\ displaystyle x = 13 {,} 543232 \ ldots | \ cdot 100}$
${\ displaystyle 100 \ cdot x = 1 {.} 354 {,} 323232 \ ldots}$
3rd step: subtract the two lines created by step 1 and 2 from each other: (the periods after the decimal point are omitted)
${\ displaystyle 10 {.} 000 \ cdot x = 135 {.} 432 {,} 323232 \ ldots}$ (Line 1)
${\ displaystyle 100 \ cdot x = 1354 {,} 323232 \ ldots}$ (Line 2)
${\ displaystyle 9 {.} 900 \ cdot x = 135 {.} 432-1 {.} 354 = 134 {.} 078 \!}$ (Row 1 minus row 2)
4th step: switch
${\ displaystyle 9 {.} 900 \ cdot x = 134 {.} 078 |: 9 {.} 900}$
Result: ${\ displaystyle x = {\ frac {134 {.} 078} {9 {.} 900}} = 13 {,} 54 {\ overline {32}}}$

#### Ambiguity of representation

A special feature of decimal fraction expansion is that many rational numbers have two different decimal fraction expansion. As described above, one can transform and make the statement ${\ displaystyle 0 {,} {\ overline {9}}}$

${\ displaystyle 0 {,} {\ overline {9}} = {\ frac {9} {9}} = 1}$

reach, see article 0.999 ...

From this identity one can further deduce that many rational numbers (namely all with finite decimal fraction expansion with the exception of 0) can be represented in two different ways: either as a finite decimal fraction with period 0, or as infinite with period 9. To make the representation unambiguous To do this, you can simply forbid period 9 (or less often period 0).

#### formula

The following formula can be set up for periodic decimal fractions with a zero in front of the decimal point:

${\ displaystyle p = {\ frac {x \ cdot \ left (10 ^ {n} -1 \ right) + y} {10 ^ {m} \ cdot \ left (10 ^ {n} -1 \ right)} }}$

Here p is the number, x is the number before the beginning of the period (as an integer), m is the number of digits before the beginning of the period, y is the sequence of digits of the period (as an integer) and n is the length of the period.

The application of this formula should be demonstrated using the last example:

${\ displaystyle p = 0 {,} 492363636 \ ldots = 0 {,} 492 {\ overline {36}}}$
${\ displaystyle x = 492; \ quad m = 3; \ quad y = 36; \ quad n = 2}$
${\ displaystyle p = {\ frac {x \ cdot \ left (10 ^ {n} -1 \ right) + y} {10 ^ {m} \ cdot \ left (10 ^ {n} -1 \ right)} } = {\ frac {492 \ cdot \ left (10 ^ {2} -1 \ right) +36} {10 ^ {3} \ cdot \ left (10 ^ {2} -1 \ right)}} = { \ frac {492 \ cdot 99 + 36} {1000 \ cdot 99}} = {\ frac {48744} {99000}} = {\ frac {677} {1375}}}$

#### period

In mathematics , the period of a decimal fraction is a digit or sequence of digits that is repeated over and over again after the decimal point. All rational numbers , and only these, have a periodic expansion of decimal fractions.

Examples:

Purely periodic: (the period begins immediately after the decimal point)
1/3 = 0 3 3333 ...
1/7 = 0, 142857 142857 ...
1/9 = 0, 1 1111 ...
Mixed periodic: (after the decimal point there is a previous period before the period begins)
2/55 = 0.0 36 363636 ... (previous period 0; period length 2)
1/30 = 0.0 3 333 ... (previous period 0; period length 1)
1/6 = 0.1 6 666 ... (previous period 1; period length 1)
134078/9900 = 13.54 32 32 ... (the previous period is 54; period length is 2)

Finite decimal fractions also count among the periodic decimal fractions; after inserting an infinite number of zeros, for example 0.12 = 0.12000 ...

Real periods (i.e. no finite decimal fractions) occur in the decimal system precisely when the denominator of the underlying fraction cannot be generated solely by the prime factors 2 and 5. 2 and 5 are the prime factors of the number 10, the base of the decimal system. If the denominator is a prime number (except 2 and 5), the period has at most a length that is one less than the value of the denominator (shown in bold in the examples). ${\ displaystyle p}$

The exact length of the period of (if the prime number is neither 2 nor 5) corresponds to the smallest natural number , wherein the prime factorization of occurring. ${\ displaystyle 1 / p}$${\ displaystyle n}$${\ displaystyle p}$${\ displaystyle R_ {n} = {10 ^ {n} -1}}$

Example for period length 6 : (10 6 - 1) = 999,999:

999,999 = 3 x 3 x 3 x 7 · 11 · 13 · 37 1/ 7 = 0 142857 142857 ... and 1/ 13 = 0, 076 923 076 923 ...

Both 1/7 and 1/13 have a period length of 6 because 7 and 13 appear for the first time in the prime factorization of 10 6 - 1. However, 1/37 has a period length of only 3, because already (10 3 - 1) = 999 = 3 3 3 37.

If the denominator is not a prime number, the period length results accordingly as the number for which the denominator is a divisor of for the first time ; the prime factors 2 and 5 of the denominator are not taken into account. ${\ displaystyle n}$${\ displaystyle R_ {n} = {10 ^ {n} -1}}$

Examples: 1/185 = 1 / (5 * 37) has the same period length as 1/37, namely 3.

1/143 = 1 / (11 13) has the period length 6, because 999.999 = 3 3 3 7 143 37 (see above)

1/260 = 1 / (2 · 2 · 5 · 13) has the same period length as 1/13, i.e. 6.

In order to determine the period length efficiently, the determination of the prime factorization of the rapidly growing number sequence 9, 99, 999, 9999 etc. can be avoided by using the equivalent relationship , i.e. repeated multiplication (starting with 1) by 10 modulo the given denominator , until this equals 1 again. For example for : ${\ displaystyle n}$${\ displaystyle 10 ^ {n} \ equiv 1 {\ pmod {k}}}$${\ displaystyle k}$${\ displaystyle k = 13}$

${\ displaystyle 10 ^ {1} \ equiv 10 {\ pmod {13}}}$,
${\ displaystyle 10 ^ {2} \ equiv 10 \ cdot 10 \ equiv 100 \ equiv 9 {\ pmod {13}}}$,
${\ displaystyle 10 ^ {3} \ equiv 9 \ cdot 10 \ equiv 90 \ equiv 12 {\ pmod {13}}}$,
${\ displaystyle 10 ^ {4} \ equiv 12 \ cdot 10 \ equiv 120 \ equiv 3 {\ pmod {13}}}$,
${\ displaystyle 10 ^ {5} \ equiv 3 \ cdot 10 \ equiv 30 \ equiv 4 {\ pmod {13}}}$,
${\ displaystyle 10 ^ {6} \ equiv 4 \ cdot 10 \ equiv 40 \ equiv 1 {\ pmod {13}}}$,

so 1/13 has the period length 6.

#### notation

For periodic decimal fractions, a notation is common in which the periodically repeated part of the decimal places is marked by an overline . examples are

• ${\ displaystyle 1/6 = 0 {,} 1 {\ bar {6}}}$,
• ${\ displaystyle 1/7 = 0 {,} {\ overline {142857}}}$.

Due to technical limitations, other conventions also exist. The overline can be placed in front, typographical emphasis (bold, italic, underlined) of the periodic part can be selected or it can be put in brackets:

• 1/6 = 0.1¯6 = 0.1 6 = 0.1 6 = 0.1 6 = 0.1 (6)
• 1/7 = 0, ¯142857 = 0, 142857 = 0, 142857 = 0, 142857 = 0, (142857)

#### Non-periodic sequence of digits after the decimal point

As explained in the article place value system , irrational numbers (also) in the decimal system have an infinite, non-periodic sequence of digits after the decimal point. Irrational numbers cannot be represented by a finite and not by a periodic sequence of digits. You can approximate with finite (or periodic) decimal fractions, but such a finite representation is never exact. So it is only possible with the help of additional symbols to indicate irrational numbers using finite representations.

Examples of such symbols are radical symbols , such as for , letters such as π or e , and mathematical expressions such as infinite series or limit values . ${\ displaystyle {\ sqrt {2}}}$

## Conversion into other place value systems

Methods for converting from and into the decimal system are described in the articles on other place value systems and under number base change and place value system .

## history

One of the oldest evidence of the decimal vorschriftlicher cultures found in a hoard of Oberding from the early Bronze Age (around 1650 v. Chr.) With 791 largely standardized clasp bars of copper from Salzburg and Slovakia. The majority of these bars had been deposited in groups of 10 by 10 bundles.

Decimal number systems - still without a place value system and without representation of the zeros - were the basis of the number scripts of the Egyptians , Minoans , Greeks and Romans, among others . These were additive number fonts, with which numbers could be written down as a memory aid when calculating , but arithmetic operations could essentially not be carried out in writing: these were rather with mental arithmetic or with other aids such as calculating stones (Greek psephoi , Latin calculi , in the late Middle Ages also called arithmetic pennies or French chips ) on the arithmetic on lines and possibly with the finger numbers .

Finger numbers according to Beda Venerabilis , left hand
Finger numbers according to Beda Venerabilis, left hand

The finger numbers, which were widespread in Roman and medieval times and also used in a slightly different form in the Arab world, were based on a decimal system for the representation of the numbers 1 to 9999, without a symbol for zero and with a position system of its own Finger positions on the left hand with the small, ring and middle finger the units 1 to 9 and with the index finger and thumb the tens 10 to 90, while on the right hand the hundreds with thumb and forefinger are a mirror image of the tens and the thousands with the three remaining fingers were shown as a mirror image of the ones. These finger numbers are said to have been used not only for counting and memorizing numbers, but also for arithmetic; The contemporary written sources, however, are limited to the description of the finger positions and do not provide any further information about the mathematical operations that can be carried out with them.

Roman handabacus (reconstruction)

On the abacus of Greco-Roman antiquity and the Christian Middle Ages, on the other hand, a fully-fledged decimal place value system was available for the representation of whole numbers, in that the number of its ones, tens, hundreds, etc. for a given number was represented by calculating stones in corresponding vertical decimal columns. On the ancient abacus, this was done by placing or pushing a corresponding number of calculi in the respective decimal column, whereby a bundling of five was also practiced, with five units each being represented by a single calculus in a special area on the side or top of the decimal column . On the monastery abacus of the early Middle Ages, which today is mostly associated with the name Gerberts and was in use from the 10th to the 12th century, the number of units in the respective decimal column was instead represented by a single stone, which was preceded by a number from 1 to 9 was numbered, while the later Middle Ages and the Early Modern Era returned to the use of un numbered arithmetic stones and the columns or now horizontally drawn lines either for decimal arithmetic with whole numbers at the base number 10 (with five bundles), or for financial arithmetic Monetary basic units inherited from the Carolingian coinage (1 libra = 20 solidi = 240 denarii) non-decimally aligned. On the ancient and medieval versions of this aid, the value zero was displayed by leaving the relevant decimal column or line blank, and so also on the monastery abacus, on which a calculating stone with an Arabic numeral (cifra) for zero was available, but was used for other purposes in the abazistic arithmetic operations. With the help of ancient and medieval abacus, addition and subtraction could be simplified considerably, while they were not very suitable for multiplication and division or required relatively complicated operations, which were described especially for the monastery abacus in medieval tracts and were notorious for their difficulty.

The Babylonians first developed a number font with a fully-fledged place value system, in which the position of the number sign also determines its value, on the basis of 60, and probably added a separate sign for zero as early as the 4th century BC. A numerical font with a place value system based on the base 10, but still without a sign for the zero, was probably created in China a few centuries before the turn of the century (attested in details since the 2nd century BC), probably with the help of chopsticks on a chessboard-like Chinese Variant of the abacus, and was only supplemented by a symbol for zero under Indian influence from the 8th century.

In India itself, the beginnings of the positional decimal system with characters for the zero cannot be determined with certainty. The older Brahmi number script , which was in use from the 3rd to the 8th centuries, used a decimal system with approaches to positional writing, but still without a zero symbol. The oldest Indian form of today's Indo-Arabic numerals , with characters for 1 to 9 and a point or small circle for zero derived from the Brahmi number script, is first outside India since the 7th century in Southeast Asia as a result of epigraphic evidence that can be reliably dated Indian export and evidence in India itself since the 9th century; however, it is believed that this numerical system began to be used in India as early as the 5th century. The same positional decimal system with signs for zero was also the basis of the almost simultaneous learned numerical system of Indian astronomers, in which circumscribing expressions such as “beginning” (1), “eyes” (2), “the three time stages” (3) for the numbers 1 to 9 and "sky", "emptiness", "point" or other words for zero were ranked according to their decimal place as a linguistic description of multi-digit numbers. Lokavibhaga , written in Prakrit in 458 , is considered early evidence of such a positional setting of in this case largely unmetaphorical linguistic numerical designations , which, however, has only survived in a later Sanskrit translation . The circumscribing numerical system is then fully developed in Bhaskara I (7th century).

For the writing of numbers, the Arabs and the peoples Arabized by them initially adopted the decimal additive system of the alphabetical Greek numerals, initially mediated by the Hebrew and Syrian models, and transferred to the 28 letters of the Arabic alphabet. By the 8th century at the latest, however, the Indian numerals and the calculation methods based on them became known first in the Arabian Orient and in the course of the 9th century in North Africa and Al-Andalus . The earliest mention is found in the 7th century by the Syrian bishop Severus Sebokht , who expressly praises the Indian system. Muhammad ibn Musa al-Chwarizmi , who not only used the new digits in his mathematical works, but also an introduction, which was only given in Latin translation, Kitāb al-Jamʿ wa- around 825, played an important role in the dissemination in the Arab and Western world . l-tafrīq bi-ḥisāb al-Hind ("About arithmetic with Indian numerals") with a description of the numerical system and the written basic arithmetic based on it that is suitable for beginners .

In 10./11. In the 19th century, Western Arabic numerals or digits derived from them ( called apices ) appeared on the calculating stones of the monastery abacus in the Latin West . But they were not also used as numerals or even for written arithmetic. Together with the monastery abacus, they were forgotten again. Al-Chwarizmi helped Indian numerals to achieve a breakthrough in Latin adaptations and related folk-language treatises since the 12th century. Their opening words Dixit Algorismi ensured that "Algorism", the Latin rendering of his name, established itself widely as the name of this new art of arithmetic. Especially in Italy, where Leonardo Fibonacci made it known in his Liber abbaci from his own knowledge acquired in North Africa, the Indian numerical calculation has been able to almost completely displace the abacus (with calculating stones) in finance and commerce since the 13th century Take names (abbaco) . In other countries it became the subject of scientific and commercial instruction, but until the early modern period it had an overpowering competitor in arithmetic on lines. Even as a simple number font for the practical purposes of writing down numbers and numbering , for which a place value system is not required, the Indo-Arabic numerals could only gradually assert themselves against the Roman numerals since the early modern period.

## literature

• John D. Barrow: Why the World is Mathematical / John D. Barrow. From the English and with an afterword by Herbert Mehrtens . Campus-Verl., Frankfurt / Main 1993, ISBN 3-593-34956-6 .
• Georges Ifrah: Universal History of Numbers. With tab. And drawing. of the author. Parkland-Verl., Cologne 1998, ISBN 3-88059-956-4 .
• Karl Menninger: number and number. Vol 2. Number Scripture. And arithmetic . Vandenhoeck & Ruprecht, 1958.
• John M. Pullan: The History of the Abacus . Hutchinson, London 1968.

Wiktionary: Decimal system  - explanations of meanings, word origins, synonyms, translations

## Individual evidence

1. Harald Haarmann: World history of numbers . Beck, Munich 2008, ISBN 978-3-406-56250-1 , pp. 29 .
3. Harald Krause, Sabrina Kutscher u. a .: Europe's largest bar stock: The early Bronze Age copper treasure from Oberding. In: Matthias Wemhoff, Michael M. Rind: Moving times: Archeology in Germany. Berlin, Petersberg 2018, p. 167 ff.
4. J. Stolz: First evidence of the decimal system? The early Bronze Age bar hoard from Oberding. In: Restauro. Journal for Conservation and Restoration , 8th year 2017, pp. 14–19.
5. Menninger: number word and number (1958), II, p. 3ff .; Karl-August Wirth, Art. Finger Numbers. In: Otto Schmidt (ed.): Reallexikon zur deutschen Kunstgeschichte , Volume VIII, Metzler Verlag, Stuttgart 1987, Sp. 1229-1310; Ifrah: Universal history of numbers (1998), p. 87.
6. Menninger: number word and number (1958), II, p. 104ff .; Ifrah: Universalgeschichte der numbers (1998), p. 136ff .; Pullan, History of the Abacus (1968), pp. 16ff.
7. Menninger: number word and number (1958), p. 131ff; Ifrah: Universalgeschichte der numbers (1998), p. 530ff .; Werner Bergmann: Innovations in the Quadrivium of the 10th and 11th Centuries. Studies on the introduction of the astrolabe and abacus in the Latin Middle Ages , Steiner Verlag, Stuttgart 1985 (= Sudhoffs Archive, supplement 26), p. 57ff., P. 174ff.
8. ^ Alfred Nagl: The arithmetic pennies and operational arithmetic. In: Numismatical Journal 19 (1887), pp. 309–368; Menninger: number and number (1958), II, pp. 140ff .; Pullan: History of the Abacus (1968), passim
9. ^ Francis P. Barnard: The Casting Counter and the Counting Board. A Chapter in the History of Numismatics and Early Arithmetic , Clarendon Press, Oxford 1916; Menninger: Number word and number (1958), II, p. 152ff., P. 165, p. 178, p. 182f .; Pullan: History of the Abacus (1968), p. 52ff .; Ifrah: Universal history of numbers (1998), p. 146ff.
10. Ifrah: Universalgeschichte der numbers (1998), p. 411ff., P. 420.
11. Ifrah: Universalgeschichte der numbers (1998), p. 428ff., P. 511ff.
12. Ifrah. Universal history of numbers (1998), p. 504ff.
13. Ifrah. Universal history of numbers (1998), p. 486ff.
14. Ifrah: Universalgeschichte der numbers (1998), p. 498ff.
15. Ifrah: Universalgeschichte der numbers (1998), p. 493ff.
16. Ifrah: Universal history of numbers (1998), p. 499f.
17. Ifrah: Universalgeschichte der numbers (1998), p. 307ff.
18. Ifrah: Universalgeschichte der numbers (1998), p. 533ff.