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There are several instances in history that are considered to be good candidates as discovery and first use of zero. Examples includes Ancient Sumarian and Babylonian civilizations, Mayans, medieval India, China and Cambodia.

What are the mathematical concepts and ideas that can be logically considered as precursors to discovery of zero. For example, familiarity and extensive use of a positional number system within a civilization could be considered as one such precursor. Another is evidence of experimenting with different symbols like empty space, dot, etc. to indicate absence of a number and need to deal with relatively large numbers.

Similarly what are the ideas that can be anticipated to follow discovery of zero? Here are some examples: negative numbers, new rules of arithmetic to handle zero and negative numbers in addition to natural numbers, etc.

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    $\begingroup$ Hi, welcome to hsm. We already have a number of threads with answers about this: Zero and positional numeral systems hsm.stackexchange.com/questions/3245/… Has a digit ever been used to represent the number “10”? hsm.stackexchange.com/questions/2181/… When was Zero actually introduced in mathematics? hsm.stackexchange.com/questions/276/… $\endgroup$ – Conifold Sep 19 '16 at 18:38
  • $\begingroup$ Thank you, Conifold, I did look and am still looking at the posts you mentioned. However my question is a little different. I would like to know what comes before and after the discovery of zero. $\endgroup$ – ramana_k Sep 19 '16 at 19:03
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    $\begingroup$ Could you explain what you are looking for more precisely? Placeholder usage is before and incorporation into algebra after, but that is well covered there and in standard histories. $\endgroup$ – Conifold Sep 19 '16 at 20:55
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The main role of zero is not to express "nothing", but to serve as a place holder in a positional system. The need in a place holder begins when people start comlicated calculations, that is in Babylonian astronomy. First they used a space instead of zero, then invented a secial cuneiform sign (which I cannot reproduce here), see Van-der-Waerden, Science Awakening.

Greeks used letter notation for numbers, but they also needed zero. Ptolemy uses the Greek letter $\omicron$ (omicron).

Thus for example the number written in modern sexagesimal notation as 373; 10,0,58 (which means 373 degrees 10 minutes 0 seconds and 58/60 of a second) in Ptolemy would look like this:

$$\overline{\tau\omicron\gamma}\;\overline{\iota}\; \omicron\; \overline{\nu\eta}$$

Horisontal overline is used to show that letters under it are digits of one number. Here $\tau$ is 300, $\gamma$ is 3, $\iota$ is 10, $\nu$ is 50 and $\eta$ is 8. The $\omicron$ has a double meaning here: overlined it stands for the 70 in 373, and not overlined for zero.

The only difference with modern notation is that Ptolemy used decimal systems for the integer part and sexagesimal for the fractional part. But "zero" he had.

Van der Waerden writes that the use of the letter $\omicron$ to denote zero was known to some Iamblichus, a Pythagorean, but this cannot be the Iamblichus of Wikipedia (245-345, a Syrian philosopher) because he lived after Ptolemy, and van der Waerden is well aware that Ptolemy used $\omicron$ as zero.

As Indians and Arabs knew Ptolemy, crediting them with "invention of zero" is an exaggeration. But of course they perfected the positional system.

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  • $\begingroup$ Ptolemy's zero looks more like an "o" with a roof over it. It looks very different from omikron = 70. $\endgroup$ – fdb Sep 19 '16 at 23:59
  • $\begingroup$ Look here: raymondm.co.uk and click on "Greek zero", for lots of examples from old manuscripts. $\endgroup$ – fdb Sep 20 '16 at 0:02
  • $\begingroup$ How did Archimedes write the value of pi? Is it something like 3.14159, but using Greek letters with a line on top? Did it change by Ptolemy's time? How about square root of 3, which has a zero in it: 1.73205 ? $\endgroup$ – ramana_k Sep 20 '16 at 4:06
  • $\begingroup$ @ThomasG. The ancients did not have decimal fractions. $\endgroup$ – fdb Sep 20 '16 at 8:27
  • $\begingroup$ Archimedes would write $\pi$ as an (approximate) simple fraction, the ratio of two integers. More precisely he would use two fractions, one slightly greater another slightly less than $\pi$. His famous approximation is 22/7. But he clearly understood that this is not an exact value. $\endgroup$ – Alexandre Eremenko Sep 21 '16 at 3:38
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in the case of Arabic the "precursors" of zero we entirely practical, not really mathematical. you need zero to balance the books; algebra emerged from the need to keep the books, esp. in cases of inheritance based on Islamic law. so zero effectively means even-steven: I don't owe you, you don't owe me, everything is fairly balanced. (The term "algebra" comes from the title of al-khwarizmi's "concise book of making whole and balancing", where "making whole" = al-jabr, literally bonesetting.)

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