A few people have already answered this but I'd like to go into a bit more detail about it (I'm a budding Mathematical Historian so this is a topic that resonates well with me.)

First, consider what mathematics would be like without no zero - obviously we have the issues such as having no additive identity, but most importantly, how would you distinguish between a number such as 56 or 506? The Babylonians dealt with this by leaving a 'space' between numbers, so the difference between 56 and 506 would be "56" and "5 6". Obviously this lends to difficulties when looking at numbers such as 5006; it isn't the easiest thing in the world to distinguish between "5 6" and "5 6". So the Babylonians eventually invented a symbol to make the difference more clear - while you could possibly consider this the first use of "zero", there's no evidence to suggest that they treated this as a proper number - so it was more of a placeholder than a number.^1

Zero was very common in Indian mathematics circa 650 CE. A stone tablet dated to 876 CE showed the use of the numbers 50 and 270 - notably, the notation for 0 is strikingly similar to the modern day notation, although their zeroes were smaller and raised. Unlike the Babylonians, however, these zeroes were being used in mathematical operations - mathematicians such as Brahmagupta, Mahavira, and Bhaskara all made use of it. Brahmagupta demonstrated that a number subtracted from itself gave 0, and any number multiplied by 0 gave 0.^2 Brahmagupta's treatise which combined the use of positive numbers, negative numbers, and zero (that is, the set of integers) was written in 628 CE.^3

The number zero was also developed independently by the Mayans circa 665 CE, but it was the development in India which influenced other people - the Arabs, Europeans and Chinese all made use of the concept.

It's possible that the Bakhsali Manuscript is the first documented use of zero as a number, but this is only because the actual date of it is uncertain. Scholars seem to believe that it was made somewhere between 200 CE and 400 CE, but no-one is certain. This also documents the use of negative numbers, with the notation used being the inclusion of a "+" after a number to indicate it as a negative.

So, to summarise, we know the Babylonians had a 'zero' as a placeholder, but they failed to see it as an actual number. It wasn't until somewhere in the region of 200-628 CE that zero was utilised as a number.

EDIT: I've removed an erroneous comment stating that negative numbers were developed around the same time as zero (although the two were used in conjunction by both Brahmagupta and the Bakhshali manuscript.)^4 Thanks to /u/mathguyhere1 for correcting me.

Sources:

^(1: Morris Kline - Mathematics in Western Culture)

^(2: Clifford Pickover - The Math Book)

^(3: Richard Elwes - Maths 1001)

^(4: Marcus du Sautoy - Finding Moonshine)

Few things to note:

• This is the first time I've done a post like this in AskHistorians so if I've made any mistakes with setting it out or the like then please let me know.

• Unfortunately I'm currently at university so a lot of books I have on this subject aren't with me right now! Apologies for only having a few sources. One of the books that I believe would be very relevant to this topic is Zero: The Biography of a Dangerous Idea.

• The Babylonians used a sexagesimal system (base 60) while we use a decimal system (base 10). I used a base 10 example when talking about the Babylonian's 'placeholder' for clarity, even if they wouldn't have written numbers anything like we do nowadays.

am i right in thinking negative numbers didn't hold nearly the same sort of conceptual problem due to the basic concept of subtraction?

hi! you may find these posts of interest

• There is no Roman numeral for the number zero. How would a Roman mathematician answer the equation X minus X or V minus V? What about V minus X?

• Did, and how did, cultures without a mathematical zero understand and represent the concept of "nothing?"

• Which was the first civilization to come up with the concept of zero?

• Why did someone have to "invent" the use of the number zero?

• How did mathematics work before the introduction of the number zero?

The oldest known description of negative numbers comes from chapter eight of the Nine Chapters on Mathematical Procedures (Jiu zhang suan shu), which is the earliest Chinese classical text that focuses exclusively on mathematics. It was first compiled during the 2nd to 1st century BCE by Zhang Cang and Geng Shoucang. The chapter is about solving systems of linear equations through the use of matrices.

The method of elimination used by Chinese mathematicians naturally led to the discovery of negative numbers. A "sign rule" for adding and subtracting negative numbers is introduced in problem 4 by Liu Hui, who wrote a commentary on the text. Negative numbers are represented by black counting rods on a counting board and positive numbers are represented by red counting rods.

Source: V.J. Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, 2007.

I highly recommend watching 'The Story of 1'. One of the best documentaries I have seen. It is all about the history of numbers. For a high school dropout who hated maths it was incredibly interesting.