Ten is special to humans, as there are 10 fingers on two hands, and fingers are still the basic counting medium for people.

So, was there any digit representing the number "10" in a positional system historically used by any civilizations?


We could have taken A as a digit to represent number 10, and thus the numbers would have been:

0   1   2   3   4   5   6   7   8   9   A
10  11  12  13  14  15  16  17  18  19  1A

Obviously, there would have been new methods of calculation.

Why was 10 not given its own digit, and instead was represented by using other digits?

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    $\begingroup$ To me, this question (as of right now) seems to either: 1) Lead to a tautological no-answer if one restricts to the signs 0-9 and a decimal numeral system, or 2) Lead to an obvious yes-answer if one broadens the scope and thinks of non-decimal numeral systems. @Barun, do you have any reason to believe otherwise? $\endgroup$ – Danu Apr 14 '15 at 15:07
  • $\begingroup$ My question seems to be confined to decimal system, which is because we use decimal system. But before the introduction of 0, how was 10 represented? $\endgroup$ – Barun Apr 14 '15 at 15:21
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    $\begingroup$ See Roman numerals : X. $\endgroup$ – Mauro ALLEGRANZA Apr 14 '15 at 15:54
  • $\begingroup$ @MauroALLEGRANZA Roman may have X to represent 10 but does not have continuity in value of digits, and digits for upto 1000. So, I want answer related to number system like Decimal. $\endgroup$ – Barun Apr 14 '15 at 16:17
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    $\begingroup$ In the original ISBN coding used for books, the check digit is based on a mod 11 calculation and thus a single symbol is needed for 10 if it turns out to be the check digit. The letter X, naturally, was used for this role of 10. $\endgroup$ – KCd Apr 15 '15 at 2:32

Yes, it has been: cuneiform 10, or more stylized Babylonian 10, the depression made by the tip of a Babylonian wedge shaped stylus on a clay tablet. When a circular stylus was used (rarely) the symbol was just $\bigcirc$. The earliest positional system was sexagesimal, with base 60, so it had cuneiform symbols for all digits from 1 to 59. Babylonians used it since before 2000 BC for commercial bookkeeping, etc. The absence of zero caused ambiguities, e.g. 1 and 60 had the same symbol. In the medial positions, however, zero was represented by a blank and later by a placeholder symbol enter image description here. Greek astronomers replaced it with $o$ in the second century BC, that could be used at the end of a number also, removing the ambiguities. They also used their letter $\iota$ for 10 instead of Babylonian cuneiform, even though it was the 9th letter of their alphabet (archaic letter $\digamma$ was used for 6).

So the use of zero as a placeholder much predates its use as a number, and Indians learned about it from Ptolemy's Almagest, if not earlier. Therefore, they would have needed no separate symbol for 10 when they switched to decimal notation. But in any case, zero is already used in 3rd century AD even as a number, while the decimal notation first appears much later, around 458 AD, so the problem never arose.

But 10 was not special among sexagesimal digits, so perhaps more in the spirit of the question is the symbol $\big|$ used in Chinese proto-decimal system before 4th century BC. That system had hieroglyphs for digits from 1 to 9, and for the powers of 10. Although digits were written in order we write them today to form a number, the symbol for a power of ten was placed above or below each, so its value wasn't indicated by position alone. This allowed for non-ambiguous representation without even placeholder 0, and the number could still be recovered if the digits got scrambled.

By the way, hexadecimal system was also used in China, mostly for calculations with weights. This was done on abacus (counting board) starting around 190 AD, and since abacus is not paper what represented 10 was not a symbol, but an arrangement of beads.

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  • $\begingroup$ Sorry, but my question is more confined to the decimal number system. $\endgroup$ – Barun Apr 15 '15 at 14:22
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    $\begingroup$ @Barun The system you describe in your edit is not decimal, it has base 11 (10 represents the base in any positional system). Methods of calculation would not be new, they are essentially the same in all positional systems. 11 or other primes weren't chosen as bases because then most used fractions (1/2,1/3,1/5...) would have infinite representations in them. In fact, 60 was originally chosen because it has so many divisors, but it also needs a lot of digits, 10 was a compromise. $\endgroup$ – Conifold Apr 15 '15 at 16:22

Posting History

It was already mentioned that the Mesopotamian (hexagezimal system) used a special symbol for our "10". Also, In the modern hexadecimal system we have A for "10". The Roman and the Greek specialties were also mentioned.

New stuff

To add something to the already existing posts I copy here a paragraph from my own lecture notes about numbers systems:

Even if these systems (Mezopotamian, the two Mayan systems and the hexadecimal system) use(d) special symbol for ten, those symbols, however, were not special like our nine in our modern decimal version of counting. Using a special symbol for ten did not make these systems of base eleven.

Base "eleven" could have been the most natural number system... To support this idea I copy here another paragraph from the same lecture notes:

symbols. That is, the number system of base eleven would have been the most natural one.

Taking into account our toes the number system of base 21 seems to be even more natural, uncomfortable though...

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  • $\begingroup$ Lovely. If I may, I'd suggest you change the wording of the second sentence "when the conquistadors discovered them" to "when the conquistadors came into contact with them" or something along those lines $\endgroup$ – chuck Mar 30 '19 at 2:21
  • $\begingroup$ @chuck: Thank you for your comment. A shall do the change. However, in this post that text is an image. So, it may take time until the change will appear. $\endgroup$ – zoli Mar 30 '19 at 11:42
  • $\begingroup$ That thing about base eleven is astonishing. $\endgroup$ – mattecapu Jul 10 '19 at 5:48

Decimal "10" = Hexadecimal "A"

Maybe you don't consider that a "digit", but we'll necessarily have to broaden the definition of "digit" to include anything that's not 0-9.

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Mauro has already mentioned the Roman numerals. The Greeks of the classical period used the letters of the alphabet to represent numbers. The first nine letters (A to Θ) stand for the units from 1 to 9, the next nine letters (I to Ϙ) stand for the multiples of ten (10, 20, 30…), the next nine (from P to ϡ) are the hundreds (100, 200…). You can write (for example) 111 as PIA. If these are not digits, then I do not know what a digit is.

In the Hellenistic period the Jews and Arabs and others adopted this system, using the letters of the Hebrew and Arabic alphabets to represent numbers in exactly the same way. This continues throughout the Middle Ages.

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Actually you can use any symbol to represent "Zero". Suppose we take it "X". But in whatever index you are working with (let us say our index is "n"). the $n^{th}$ number in that index has to be (#symbol for "1")(symbol for "0").

For example if your arithmetic is based on index of "7" and we use

"A" for "1"

"B" for "2"

"C" for "3"

"D" for "4"

"E" for "5"

"F" for "6"

And say "X" for "0".

then seventh number (equivalent to "10" in deimal arithmetic) will be "AX".

This answers why "ten" is represented just by two digits "1" and "0". in a given order?

Chek that any other representation say "8""9" will collapse whole arithmetic!

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  • $\begingroup$ This is an interesting idea, but it isn't based off of a specific historical system, which is what we're looking for. $\endgroup$ – HDE 226868 Apr 19 '15 at 17:56

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