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In the history of numbers, negative numbers as well as zero appear relatively late, possibly because the concepts represented are not really 'quantities' in a straightforward sense. However, even between these two, in many cultures $0$ seems to have been introduced as a number before the advent of negative numbers.

Question: Do you know of any culture that had negative numbers before they had zero?

Some thoughts of mine:

In the group theoretic formulation of arithmetic the concept of inverses doesn't even make sense without the notion of a neutral element. And for something to be a number, one should be able to calculate with it. One might therefore argue that if a culture had some negative number $-a$, they would need to have zero, because they would need some rule to add $-a$ and $a$.

However, the concept of negative numbers could have been more familiar because of financial debts (for instance), without there having to be a 'numerical' notion of zero.

The wikipedia page on negative numbers contains some information, but nothing conclusive. I am also aware that it is not perfectly clear what is meant here by 'number', but this should not prevent an answer to the question. Thank you.

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Well the Chinese were the first to successfully represent positive and negative numbers and perform arithmetic with such numbers without having an explicit number 0 (although they did have the concept of zero, they had no symbol for it). They used a counting rod system where black rods represented negative numbers and red ones represented positive numbers.

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But the rod counting was done on a surface ruled into squares, and one could leave a square empty to represent zero. So in that system there was a digit for zero, namely an empty square, and a numeral zero, namely a row of empty squares. (More details) – MJD Oct 20 '12 at 15:32

There is one obvious number system which has negative numbers but not zero, and that is the system that we use to number our years (AD and BC, or CE and BCE). Years are counted from some starting point using the ordinal numbers first, second, etc. (as opposed to the cardinal numbers one, two, etc.). Earlier events are given a position in time by counting backwards from the same starting point. There is no year zero, so the year 1 AD is immediately preceded by 1 BC.

The first person to use this system for events before the starting point seems to be the venerable Bede, in the early 8th century. Negative numbers where not in common use in Europe at the time, but, according to Wikipedia, Bede did use the number zero in his Easter calculations.

This number system is of course not isomorphic to the integers, but I think it is nevertheless reasonable to call it a number system with positive and negative numbers. The direction one is counting in is designated by some symbol (AD or BC, which could as well be $+$ or $-$), and there is a symmetry so that the years $x$ AD and $x$ BC are equidistant from the starting point.

An example which is isomorphic to the number system above is the American convention for numbering floors of a building with a basement.

For obvious reasons, such a number system is only suitable for counting, not for doing calculations. As the examples above show, different number systems may coexist at the same time within one culture. Hence, it is possible for other cultures to have used a similar number system for specialized purposes, while possibly using number systems with zero for other tasks. I don't know of any examples before Bede, though.

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