Base 10

The basic idea of this system is that of grouping the symbols (in their regular sequence) in "packets" of tens, hundreds (tens of tens), thousands (tens of tens of tens), and so on.
Obviously, instead of using threaded shells and leather straps, we could apply the same system to words or to graphic signs, producing oral or written decimal numeration.
Our current number system is just such, using the following graphic signs, often referred to as Arabic numerals:

1 2 3 4 5 6 7 8 9 0

The first nine symbols represent the simple units, or units of the first decimal order (or "first magnitude"). They are subject to the rule of position, or place-value, since their value depends on the place or position that they occupy in a written numerical expression (a 5, for instance, counts for five units, five tens, or five hundreds depending on its position in a three-digit numerical expression).
The tenth symbol above represents what we call "zero", and it serves to indicate the absence of any unit of a particular decimal order, or order of magnitude, It also has the meaning of "nought" - for example, the number you obtain when you subtract a number from itself.
The base of ten, which is the first number that can be represented by two figures, is written as 10, a notation which means "one ten and no units".
The numbers from 11 to 99 are represented by combinations of two of the figures according to the rule of position:

11 "one ten, one unit"
12 "one ten, two units"
20 "two tens, no units"
21 " two tens, one unit"
30 "three tens, no units"
40 "four tens, no units"
50 "five tens, no units"

The hundred, equal to the square of the base, is written: 100, meaning "one hundred, no tens, no units" and is the smallest number that can be written with three figures.
Numbers from 101 to 999 are represented by combinations of three of the basic figures:

101 "one hundred, no tens, one unit"
102 "one hundred, no tens, two units"
...
753 "seven hundreds, five tens, three units"
...
998 "nine hundreds, nine tens, eight units"
999 "nine hundreds, nine tens, nine units"

There then comes the thousand, equal to the cube of the base, which is written: 1000 ("one thousand, no hundreds, no tens, no units"), and is the smallest number that can be written with four figures. The following step on the ladder is the ten thousand, the base to the power of four, which is written 10000 ("one ten thousand, no thousands, no hundreds, no tens, no units") and is the smallest number that can be written with five figures; and so on.

Numbers in English

In his work Universal history of numbers Ifrah reports a complete example. We report it in the following.
In English the first ten numbers have individual name:

1	2	3	4	5	6	7	8	9
one	two	three	four	five	six	seven	eight	nine

The first nine are "units of the first decimal order of magnitude" and the tenth constitutes the "base" of the system (and by definition is therefore the sign for the "unit of the second decimal order of magnitude").
To name the numbers from 11 to 19, the units are grouped in "packets" and we proceed by simple addition:

11 one-ten (=1+10)
12 two-ten (=2+10)
13 three-ten (=3+10)
14 four-ten (=4+10)
15 five-ten (=5+10)
16 SIx-ten (=6+10)
17 seven-ten (=7+10)
18 eight-ten (=8+10)
19 nine-ten (=9+10)

Multiples of the base, from 20 to 90, are the "tens", or units of the second decimal order, and they are expressed by multiplication:

20 two-tens (=2x10)
30 three-tens (=3x10)
40 four-tens (=4x10)
50 five-tens (=5x10)
60 six-tens (=6x10)
70 seven-tens (=7x10)
80 eight-tens (=8x10)
90 nine-tens (=9x10)

If the number of tens is itself equal to or higher than ten, then the tens are also grouped in packets of ten, constituting the "units of the third decimal order", as follows:

100 hundred
200 two hundreds
300 three hundreds
...

The hundreds are themselves then grouped into packets of ten, constituting "units of the fourth decimal order", or thousands:

1000 one thousand (=10³)
2000 two thousands (2x1000)
3000 three thousands (3x1000)
...

Then come the ten thousands, which used to be called myriads, corresponding to the "units of the fifth decimal order":

10000 a myriad (=10⁴)
20000 two myriads (2x10000)
30000 two myriads (3x10000)
...

More in general, the names of all the other numbers are obtained by creating expressions that rely simultaneously on multiplication and addition in strict descending order of the powers of the base 10:

53781

five-myriads	three-thousands	seven hundreds	eight-tens	one
5x10000	+3x1000	+7x100	+8x10	+1

It must have taken a very long time for people to develop such an effective way of naming numbers, as it obviously presupposes great powers of abstraction

Particular oral traditions and the rules of individual languages produce a wide variety of irregularities into the theoretical naming system: in the following are some characteristic examples from around the world.

Numbers in Tibetan

Also Tibetan has an individual name for each of the first ten numbers:

1	2	3	4	5	6	7	8	9	10
gcig	gnyis	gsum	bzhi	lnga	drug	bdun	brgyad	dgy	bcu

For numbers from 11 to 19, Tibetan uses addition, just like seen above for English numbers:

11 bcu-gcig (=10+1)
12 bcu-gnyis (=10+2)
13 bcu-gsum (=10+3)
14 bcu-bzhi (=10+4)
15 bcu-lnga (=10+5)
16 bcu-drug (=10+6)
17 bcu-bdun (=10+7)
18 bcu-brgyad (=10+8)
19 bcu-dgu (=10+9)

And for the tens, multiplication is applied (also in this case just like as seen above):

20 gnyis-bcu "two-tens” (=2x10)
30 gsum-bcu "three-tens (=3x10)
40 bzhi-bcu "four-tens" (=4x1O)
50 Inga-bcu "five-tens" (=5x10)
60 drug-bcu "six-tens" (=6x10)
70 bdun-bcu "seven-tens" (=7x10)
80 brgyad-bcu "eight-tens" (=8x10)
90 dgu-bcu "nine-tens" (=9x10)

For a hundred (=10²) there is the word brgya, and the corresponding multiples are obtained by the same principle of multiplication: for the tens, multiplication is applied:

200 gnyis-brgya "two-hundreds" (=2x100)
300 gsum-brgya "three-hundreds"(=3x100)
400 bzhi-brgya "four-hundreds" (=4x100)
500 Inga-brgya "five-hundreds" (=5x100)
600 drug-brgya "six-hundreds" (=6x100)
700 bdun-brgya " seven-hundreds" (=7x100)
800 brgyad-brgya "eight-hundreds" (=8x100)
900 dgu-brgya "nine-hundreds" (=9x100)

There are similarly individual words for "thousand", "ten thousand" and so on, producing a very simple naming system for all intermediate numbers:

21 gnyis-bcu rtsa gcig "two-tens and one" (=2x10+1)
560 lnga-brgya rsta drug-bcu "five-hundreds and six-tens" (=5x100+6x10)

Numbers in Mongolian

Numbering in Mongolian is similarly decimal, but with some variations on the regular system we have seen in Tibetan. It has the following names for the first ten numbers:

1	2	3	4	5	6	7	8	9	10
nigän	qoyar	gurban	dörbän	tabun	jirgu’an	dolo’an	naiman	yisün	arban

and proceeds in a perfectly normal way for the numbers from eleven to nineteen:

11 arban nigän ("ten-one")
12 arban qoyar ("ten-fwo")
...

The tens are formed rather differently. Instead of using analytic combinations of the "two-tens", "three-tens" type, Mongolian has specific words formed from the names of the corresponding units. subjected to a kind of declension of the ending of the word:

20 qorin (from qoyar = 2)
30 gučin (from gurban = 3)
40 döčin (from dörbän = 4)
50 tabin (from tabun = 5)
60 jirin (from jirgu’an = 6)
70 dalan (from dolo’an = 7)
80 nayan (from naiman = 8)
90 jarin (from yisün = 9)

From one hundred, however, numbers are formed in a regular way based on multiplication and addition, as explained above:

100 ja'un ("hundred")
200 qoyar ja'un ("two-hundreds")
300 gurban ja'un ("three-hundreds")
...
1000 minggan ("thousand")
2000 qoyar minggan ("two-thousands")
3000 gurban minggan ("three-thousands")
...
10000 tümän ("myriad")
20000 qoyar tümän ("two-myriads")
...

20541	qoyar tümän		tabun ja'un	döcin	nigän
	two-myriads		five hundreds	forty	one
	2x10000	+0x1000	+5x100	+40	+1

Numbers in Ancient Turkish

The first Ancient Turkish numbers are as follows:

1	2	3	4	5	6	7	8	9	10
bir	iki	üč	tört	beš	altï	yěti	säkiz	tokuz	on

For the tens, the following set of names are used:

10 on
20 yegirmi
30 otuz
40 kïrk
50 ällig
60 altimïš
70 yětmiš
80 säkiz on
90 tokuz on

The tens from 20 to 50 do not seem to have any etymological relation with the corresponding units, however, the word for 50 is very probably derived from the ancient method of finger-counting, since ällig is clearly related to äliii (or älig), the Turkish word for "hand". The system then gives the special name of yüz to the number 100, and proceeds by multiplication for the names of the corresponding multiples of a hundred:

100 yüz
200 iki yüz "two-hundreds” (=2x100)
300 otuz "three hundreds" (=3x200)
400 kïrk "four-hundreds" (=4x100)
500 ällig “five-hundreds" (=5x100)
600 altimïš "six-hundreds" (=6x100)
700 yětmiš "seven hundreds" (=7x100)
800 säkiz on "eight-hundreds" (=8x100)
900 tokuz on "nine-hundreds" (=9x100)

The word for a thousand is bïng (which in some Turkic dialects also means "a very large amount"), and the multiples of a thousand are similarly expressed by analytical combinations of the same type:

1000 bïng
2000 iki bïng "two-thousands" (=2x1000)
3000 üč bïng "three-thousands" (=3x1000)
...

What is unusual about this system is the way the numbers from 11 to 99 are expressed. In this range, what is given is first the unit, and then, not the multiple of ten already counted, but the multiple not yet reached. What is involved is neither a multiplicative nor a subtractive principle but something like an ordinal device. For example:

11 bir yegirmi literally: "one, twenty" meant as “the first unit before twenty”
12 iki yegirmi ("two, twenty" meant as "the second unit before twenty")
13 üč yegirmi ("three, twenty" meant as "the third unit before twenty")
21 bir otuz ("one, thirty" meant as "the first unit before thirty")
22 iki otuz ("two, thirty" meant as "the second unit before thirty")
78 säkiz säkiz on ("eight, eighty" meant as "the eighth unit before eighty")
99 tokuz yüz ("nine, one hundred" meant as "the ninth unit before a hundred")

This way of counting is reminiscent of the way time is expressed in contemporary German, where, for "a quarter past nine" you say viertel zehn, meaning "a quarter of ten" (= "the first quarter before ten"), or for "half past eight" you say halb neun, meaning "half nine" (= "the first half before nine").
However, around the tenth century AD, under Chinese influence, which was very strong in the eastern Turkish-speaking areas, this rather special way of counting was "rationalised". Using the Turkish stem artuk, meaning "overtaken by", the following expressions were created:

11 on artukï bir ("ten overtaken by one")
23 yegirmi artukï üč ("twenty overtaken by three")
53 ällig artukï üč ("fifty overtaken by three")
87 säkiz on artukï yeti ("eighty overtaken by seven")

Whence come the more modern simplified versions:

11 on bir (=10+1)
23 yegirmi üč (=20+3)
53 ällig üč (=50+3)
87 säkiz on yeti (=80+7)

Numbers in Sanskrit

The numbering system of Sanskrit, the classical language of northern India, is of great importance for several related reasons.
First of all, the most ancient written texts that we have of an Indo-European language are the Vedas (Sanskrit वेदाः véda, "knowledge"), a large body of texts written in Vedic Sanskrit, from around the fifth century BC, but with traces going as far back as the second millennium BC.
The Indo-European languages are a family of several hundred related languages and dialects.
Indo-European languages are spoken by almost three billion native speakers, the largest number by far for any recognised language family. Of the twenty languages with the largest numbers of native speakers according to SIL Ethnologue, twelve are Indo-European: Spanish, English, Hindi, Portuguese, Bengali, Russian, German, Marathi, French, Italian, Punjabi, and Urdu, accounting for over 1.7 billion native speakers. Pratically all modern European languages with the exceptions of Finnish, Hungarian, Basque, and Turkish belong to the IndO-European group.
Secondly, Sanskrit, as the sacred language of Brahmanism (Hinduism), was used throughout India and Southeast Asia as a language of literary and scholarly expression, and (rather like Latin in mediaeval Europe) provided a means of communication between scholars belonging to communities and lands speaking widely different languages.
The numbering system of Sanskrit, as a part of a written language of great sophistication and precision, played a fundamental role in the development of the sciences in India, and notably in the evolution of a place-value system.
The first ten numbers in Sanskrit are as follows:

1	2	3	4	5	6	7	8	9	10
eka	dvau, dva, dve, dvi	trayas, tisras, tri	catvaras, catasras, catvari, catrur	pañca	sat	sapta	Aṣṭau, aṣṭa	náva	daśa

Numbers from 11 to 19 are then formed by juxtaposing the number of units and the number 10:

11 eka-daśa "one-ten" (=1+10)
12 dva-daśa "two-ten" (=2+10)
13 tri-daśa "three-ten" (=3+10)
14 catvari-daśa "four-ten" (=4+10)
15 pañca-daśa "five-ten" (=5+10)
16 sat-daśa "six-ten" (=6+10)
17 sapta-daśa "seven-ten" (=7+10)
18 aṣṭa-daśa "eight-ten" (=8+10)
19 náva-daśa "nine-ten" (=9+10)

For the following multiples of 10, Sanskrit has names with particular features:

20 viṃśati
30 triṃśati
40 catváriṃśati
50 pañcáśat
60 ṣaṣti
70 sapti
80 aṣiṭi
90 návati

Broadly speaking, the names of the tens from 20 upwards are formed from a word derived from the name of the corresponding unit plus a form for the word for 10 in the plural.
One hundred is śatam or śata, and for multiples of 100 the regular formula is used:

100 śatam, śata
200 dviśata (=2x100)
300 triśata (=3x100)
400 caturśata (=4x100)
500 pañcasata (=5x100)

For 1000, the word sahásram or sahásra is used, in analytical combination with the names of the units, tens and hundreds to form multiples of the thousands, the ten thousands, and hundred thousands:

1000 sahásra
2000 dvisahásra (=2x1000)
3000 trisahásra (=3x1000)
…
10000 daśasahásra (=10x1000)
20000 viṃśatsahásra (=20x1000)
30000 triṃśatsahásra (=30x1000)
…
1-00000 śatsahásra (=100X1000)
200000 dviśatsahásra (=200X1000)
300000 triśatsahásra (=300X1000)

This gives the following expressions for intermediate numbers:

4769

9	+60	+7x100	+	4x1000
nava	ṣaṣti	saptaśata	ca	catursahásra
nine	sixty	seven-hundreds	and	four-thousands

Sanskrit thus has a decimal numbering system, like ours, but with combinations done "in reverse", that is to say starting with the units and then in ascending order of the powers of 10.

Number Words of Some Indo-European Languages Showing The Extraordinary Stability of Number Words

	Sanskrit	Ancient Greek	Latin	Italian	French	German	English	Russian
1	eka	en	unus	uno	un	eins	one	odyn
2	dva	duo	duo	due	deux	zwei	two	dva
3	tri	tri	tres	tre	trois	drei	three	tri
4	catur	tetra	quatuor	quattro	quatre	vier	four	chetyre
5	panca	pente	quinque	cinque	cinq	fünf	five	piat
6	sas	hex	sex	sei	six	sechs	six	shest
7	sapta	hepta	septem	sette	sept	sieben	seven	sem
8	asta	octo	octo	otto	huit	acht	eight	vosem
9	nava	ennea	novem	nove	neuf	neun	nine	deviat
10	daca	deca	decem	dieci	dix	zehn	ten	desiat
100	cata	ecaton	centum	cento	cent	hundert	hundred	sto
1000	sehastre	xilia	mille	mille	mille	tausend	thousand	tysiaca

Number The Language of Science - by Dantzig T., Mazur J. - Pearson 2005

The Universal History of Numbers: From Prehistory to the Invention of the Computer - by Georges Ifrah - Wiley 2000