It is well known that positional numeral systems are not possible without the concept of zero and corresponding notation. This is the necessary condition, but not sufficient one. My question is:

Are there any ancient civilizations which had concept of zero but didn't not use positional numerals for any somewhat non-negligible (from historical point of view) amount of time?

If there are such civilizations, why didn't they adopt/come up with positional numeral system shortly after introducing zero?


3 Answers 3


The concept of zero (as a number) is not necessary for positional notation, one can do with a placeholder symbol, used like a punctuation mark, or even with just a blank space. This is exactly what Sumerians and Babylonians did in their sexagesimal system of numerals since 3rd millenium BC, the earliest known positional system. This system was later transmitted to Hellenistic Greece in late 3rd or second century BC, and was adopted by Greek astronomers including Hipparchus for their calculations. Ptolemy uses the placeholder symbol in his Almagest, it looks like letter o (one speculation is that it stood for "obol", a coin of smallest value). But as MacTutor in its History of Zero writes "one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself". You can see placeholders in the sexagesimal and some other historical positional systems in Has a digit ever been used to represent the number "10"?

Indians most likely encountered the sexagesimal system in their contacts with the Hellenistic world, although there is no direct evidence of transmission. Around 500 AD Aryabhata used a decimal positional system without even a placeholder zero, and some other Indian manuscripts used a dot as a placeholder. Aryabhata's word "kha" for the blank will later be used as the name for zero. But by the time Brahmagupta started talking about zero as a number over a century later positional notation was already established for millenia. I am not aware of any place where the development went the other way, from number zero to positional notation. Zero number appears to be a far more challenging concept than a punctuation mark.

  • $\begingroup$ I'd also poin out that Romans used the latin word nulla, alongside their numerals. $\endgroup$
    – user10930
    Jan 18, 2020 at 6:21
  • $\begingroup$ Conifold, Was the Brahmagupta's "number zero" equal to the "placeholder zero"? It's not clear for me if Brahmagupta creates the "algebraic zero" (a new entity) or do operations with the "placeholder zero" (a known entity). $\endgroup$
    – Pedro
    May 24, 2021 at 15:52
  • $\begingroup$ @Pedro Once you individuate it enough to start doing algebra on it it is no longer a placeholder. $\endgroup$
    – Conifold
    May 24, 2021 at 16:37
  • $\begingroup$ Note that digit position can be marked by any symbol or even "just a blank space", and so the circled blank space used by the Hindus and Ptolemy ("0") is a less error prone way of calling out and marking the absence of a (nonzero) digit in a digit position. $\endgroup$
    – mgkrebbs
    Jul 5, 2021 at 20:50

According to wikipedia article on the Number 0, by 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.

Egyptians did not used a positional numeration system before its introduction by the Persian mathematician Al Khwarizmi in 813 AD.

  • $\begingroup$ Wasn't Ptolemy an Egyptian? He used base-60 positional notation, as did the Babylonians long before him. $\endgroup$
    – fdb
    Sep 20, 2016 at 9:55
  • 1
    $\begingroup$ Ptolemy lived in the 2nd century AD. I am talking about 2 millenia before.<br> $\endgroup$
    – AlainD
    Sep 22, 2016 at 7:16
  • 1
    $\begingroup$ Ptolemy lived in the 2nd century AD. I was talking about a 2 millenia before. However, Ptolemy's mathematics was "greek" rathetr than "egyptian". He used Grek Numerals, which is positional for number since 60, but additive up to 60. $\endgroup$
    – AlainD
    Sep 22, 2016 at 7:21

It is well known that positional numeral systems are not possible without the concept of zero and corresponding notation. This is the necessary condition, but not sufficient one.

This is wrong. See eg Wikipeda:

The bijective base-10 system is a base ten positional numeral system that does not use a digit to represent zero. It instead has a digit to represent ten, such as A.


In the bijective base-26 system one may use the Latin alphabet letters "A" to "Z" to represent the 26 digit values one to twenty-six. (A=1, B=2, C=3, ..., Z=26)

With this choice of notation, the number sequence (starting from 1) begins A, B, C, ..., X, Y, Z, AA, AB, AC, ..., AX, AY, AZ, BA, BB, BC, ...

Each digit position represents a power of twenty-six, so for example, the numeral ABC represents the value 1 × 262 + 2 × 261 + 3 × 260 = 731 in base 10.

Many spreadsheets including Microsoft Excel use this system to assign labels to the columns of a spreadsheet, starting A, B, C, ..., Z, AA, AB, ..., AZ, BA, ..., ZZ, AAA, etc.

And for your question about history:

Forslund (1995) appears to be another rediscovery, and hypothesizes that if ancient numeration systems used bijective base-k, they might not be recognized as such in archaeological documents, due to general unfamiliarity with this system.

(Emphasis added.)


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