## India, China, Zero and the Negative Numbers

• 4k
I asked this on a math forum but nobody could/would give a satisfactory answer. I'm hoping this forum will answer it for me.

According mathematicians, zero, the number, was discovered by the great Indian mathematician Brahmagupta (598 - 668 AD) and negative numbers by Chinese mathematician Liu Hiu (225 - 295 AD).

Mathematicians also believe that the Chinese had no concept of zero.

My question is how can it be that the Chinese knew about negative numbers, defined as numbers less than zero, and didn't know about zero itself?

Is this there a logical explanation for this?

I mean there's a contradiction if you think the Chinese knew about negative numbers but had no concept of zero because negative numbers are defined as numbers less than zero.

• 8.6k
I know one bit of trivia, which is the symbol 0 is taken from the hole in the middle seat of a dhow where the mast went. Kind of figures that it should be a hole.
• 4k
I know one bit of trivia, which is the symbol 0 is taken from the hole in the middle seat of a dhow where the mast went. Kind of figures that it should be a hole.

Thanks for the tidbit. I actually find no logical connection between a dhow and mathematicians. Why would a dhow interest a mathematician? I'm genuinely interested. Since numbers back then were used mostly in trade/finance I think the dhow connection makes some sense since dhows were used for trade. Care to make an educated guess? I'm all ears.
• 547
The digit 'zero' and the concept of 'zero' are two different ball games.
The first, like negative numbers, is merely a place marker on an axis denoting equal interval measurement'. The second can involve all sorts of mental scenarios like 'a set with no members' or the philosophical issue of what nothing means.
Note that 'numbers' are already abstract concepts with potential applications to counting and measuring. Students who can't get their heads round elementary algebra, don't realize that it is just a second level of abstraction, numbers being the first.
• 4.2k
If you don't have zero than a negative number is simply a number less than one. No contradiction there.
• 8.6k
Why would a dhow interest a mathematician?

I think it reflects the way in which symbols were drawn from articles in everyday life, although the symbolic resonance between a hole made as a place for something and the idea of zero is interesting. I have an idea it’s also the source of the original term for ‘zero’, although must admit I’ve forgotten where I read this - might have been a review of Charles Seife’s book on the concept of zero.
• 4k

Are you trying to say that the Chinese had the concept of zero as a number but just didn't have a symbol for it?

If you don't have zero than a negative number is simply a number less than one. No contradiction there.

Although the timeline is a bit fuzzy the Chinese knew about fractions which can be less than one. If they defined negative numbers simply as less than one they wouldn't have been able to distinguish fractions from negative numbers and they were clearly able to do that.

In the Han Dynasty, the Chinese made substantial progress on root extraction and linear algebra.[2] The major texts from the period, The Nine Chapters on the Mathematical Art and the Writings on Reckoning gave detailed processes to solving mathematical problems in daily life.[3] All procedures were computed using a counting board in both texts, and they included negative numbers as well as FRACTIONS. — Wikipedia
(Emphasis mine)
• 4k
:up: :ok:
• 547
I have no idea about Chinese concepts. Its origins as a place marker symbol in a number system I think dates back to the Sumerians 3rd Century BC. This may not have been the symbol '0'. Your use of the word 'discovered' above is questionable....maybe 'established a useful concept of' , would be more accurate.
• 4.2k
Although the timeline is a bit fuzzy the Chinese knew about fractions which can be less than one. If they defined negative numbers simply as less than one they wouldn't have been able to distinguish fractions from negative numbers and they were clearly able to do that.

Fair enough. You still don't need zero to distinguish between positive and negative numbers though. What matters is use, application. You use positive numbers when you collect stuff, you use negative numbers when you remove stuff, or try to consider how much more stuff you need to meet your requirements. Or somesuch. Definitions are a mathematicians plaything.
• 728
My question is how can it be that the Chinese knew about negative numbers, defined as numbers less than zero, and didn't know about zero itself?

According to Wikipedia they did know about zero but just lacked a symbol for it.

Red rods represent positive numbers and black rods represent negative numbers.[7] Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter.

As an example of how that might work, suppose you have $2 of assets and$5 of debt. In determining your net financial situation, you note that a dollar of debt negates a dollar of assets. So this can be represented with a row of 2 red rods and a row of 5 black rods. Removing 2 red rods and 2 black rods leaves 3 black rods (and a blank space where the red rods were). Thus, in effect, you have $3 of debt and no assets. • 4k I don't know what sort of problems the Chinese were solving when the encountered negative numbers so that's a dead-end for this discussion. However there's a necessary relationship between zero and negative numbers: Brahmagupta's Brahmasphuṭasiddhanta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers.[20] The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right... — Wikipedia Brahmagupta was the mathematician who was the first to lay down the rules for calculating with zero and negative numbers. This is a convincing piece of evidence that zero and negative numbers are so closely associated that discovering one implies that you knew about the other. Your use of the word 'discovered' above is questionable....maybe 'established a useful concept of' , would be more accurate. I'm referring to the stage in mathematical thinking where actual operations are performed on negative numbers and zero. I agree that zero as a placeholder had much older origins but then it wasn't a number in it's own right - operations on zero were still undefined. That makes sense. Thanks. • 728 I don't know what sort of problems the Chinese were solving when the encountered negative numbers so that's a dead-end for this discussion. It's a fascinating topic. For the details, see Solving a System of Linear Equations Using Ancient Chinese Methods by Mary Flagg. Two quotes from that paper: "The Nine Chapters is a series of 246 problems and their solutions organized into nine chapters by topic. The topics indicate that the text was meant for addressing the practical needs of government, commerce and engineering." "[ I] read the Nine Chapters as a boy, and studied it in full detail when I was older. [ I] observed the division between the dual natures of Yin and Yang [the positive and negative aspects] which sum up the fundamentals of mathematics." - Liu Hiu (the third century mathematician) -- So the Cartesian number line and the Chinese red and black rod system are different ways to conceptualize negative numbers. In the former, a negative number is less than zero. In the latter, positive and negative numbers are duals. "Nothing" (zero) can either be the absence of a number or the secondary consequence of duals canceling, such as with a$2 sale and a \$2 purchase.

To illustrate, here's one problem from The Nine Chapters on the Mathematical Art:

Problem 8: Now sell 2 cattle and 5 sheep to buy 13 pigs. Surplus 1000 cash. Sell 3 cattle and 3 pigs to buy 9 sheep. There is exactly enough cash. Sell 6 sheep and 8 pigs, then buy 5 cattle. There is 600 coins deficit. Tell: what is the price of a cow, a sheep and a pig, respectively?

Note that the Chinese considered selling as positive and buying as negative. So the 2 sold cattle would be represented by 2 red rods, the 5 bought cattle would be represented by 5 black rods and "exactly enough cash" would be represented by a blank space.

This problem is on p10 of the linked paper, the answer on p15 and the ancient Chinese array along with the modern matrix representations are on p30.
• 4k
:up: Thanks :ok:
• 811
I actually find no logical connection between a dhow and mathematicians. Why would a dhow interest a mathematician? I'm genuinely interested.

Can't answer that specifically but there are other examples of the same thing. The Latin word for pebble is calculi, which gives us calculation. When we calculate we are literally pushing pebbles around. Even if we we're pushing them really fast through semiconductor circuits, logically we are only pushing around pebbles. A lot of technical math terms have their root in non-related or metaphorical objects. In higher math they have sheaves and germs and stalks, evoking their meaning in nature.
• 702
My question is how can it be that the Chinese knew about negative numbers, defined as numbers less than zero, and didn't know about zero itself?

Let me offer a completely unjustified speculation on that.

As soon as people started farming, they ended up with periodical harvests and a problem of warehousing stocks of agricultural produce. You also had to protect the fields, the harvests, and the stocks. So, specialization kicked in, with some people getting paid to beat the hell out of the occasional scavenging tribes, which would otherwise confiscate the inventory of produce.

You do have an "add 15 bags" transaction in such inventory, while you also have a "substract 10 bags" one. So, "add 15" could be abbreviated as "A15", while "substract 10" as "S10".

You really don't need to represent them as "+15" and "-10", because that would defeat the object of explaining what happened:

Between the 4th millennium BC and the 3rd millennium BC, the ruling leaders and priests in ancient Iran had people oversee financial matters. In Godin Tepe (گدین تپه) and Tepe Yahya (تپه يحيی), cylindrical tokens that were used for bookkeeping on clay scripts were found in buildings that had large rooms for storage of crops. In Godin Tepe's findings, the scripts only contained tables with figures, while in Tepe Yahya's findings, the scripts also contained graphical representations.[4] The invention of a form of bookkeeping using clay tokens represented a huge cognitive leap for mankind.[5]

These people quickly discovered that "A15" + "S10" = "A5".

So, several transactions in both directions can be represented by one aggregate transaction that will have one particular direction. In this context, there is absolutely nothing special about "S10" (or "-10") because it naturally emerges out of the fray.

However, these people were not interested whatsoever in adding a transaction with no items added or removed. Hence, "A0" or "S0" did not occur in their books. Since it reflects that nothing happened, why would they record a non-event? Hence, there is no need for representing transactions with a zero magnitude. There is also no need to aggregate them.

Of course, you could still occasionally end up with a zero balance, but again, you can just report the natural-language term "nothing" in that case. This "nothing" does not need to participate in aggregating calculations. Just do not enter "nothing" in such calculations, and the totals will still be correct. Adding zero to a number does not change that number. So, "not adding" zero does not change it either.

You do not need zero for adding or substracting numbers because you can just cross out the zero and the sum will still be correct. Hence, for the financial management of their inventories they were not interested in expressions like "5+0=5". It was irrelevant.

Well, that is my speculative take on why they could not be bothered to calculate with the number zero, back then.

It is not that the language expression "nothing" did not exist. I am quite sure that it did. It is also not that they did not know that adding nothing to three will yield three as a result. It is just that formalizing all of that was not needed for the basic accounting of warehouses full of wheat or rice.
• 4k
Thank you. Sorry for being lazy about this but do you have an idea about what kind of problems Brahmagupta was dealing with when he needed to formalize zero/nothing?

I mean when does nothing quantified as zero become necessary?

Personally I think zero began simply as a symbol for nothing and the rules of mathematical operations were a later development. I have no idea what the actual problems were in which zero was used as a number and not just a symbol for nothing.
• 702
Personally I think zero began simply as a symbol for nothing and the rules of mathematical operations were a later development. I have no idea what the actual problems were in which zero was used as a number and not just a symbol for nothing.

In my impression, the systematic use of the digit zero became a necessity with the introduction of the decimal (positional) system. For example, the number 504 has a zero in the middle, because of the mere bureaucratic-administrative formalisms imposed by the decimal place system. It only has a syntactic meaning.

Still, that system makes arithmetic easier. The procedures for addition, substraction, multiplication, and division are incredibly straightforward in comparison to the ancient, Roman numerals.

But then again, when this system was new, people could already calculate with Roman numerals, and were undoubtedly good at it. So, the new system did not solve a problem. On the contrary, it created one! Now they had to learn to do something in a different way, while they already knew how to do it in the old way and get perfect results.

Although Al-Khwarzimi also wrote a book about Hindu arithmetic in 825, his Arabic original was lost, and only a 12th-century translation is extant.[1] Kushyar ibn Labban did not mention the Indian sources for Hindu Reckoning, and there is no earlier Indian book extant which covers the same topics as discussed in this book.

As you can see, pretty much nobody wanted to read that book. That is undoubtedly why all copies were lost.

Leonardo Fibonacci brought this system to Europe, his book Liber Abaci introduced Arabic numerals, the use of zero, and the decimal place system to the Latin world. The numeral system came to be called "Arabic" by the Europeans, it was used in European mathematics from the 12th century, and entered common use from the 15th century to replace Roman numerals.[13][14]

So, after introducing it, it took another 300 years for people to switch to it.

You see, if you already know how to do arithmetic with roman numerals, then you will most likely think that the new system is just bullshit.

All the existing registers are done in roman numerals. All the accounting books are done in them. Why would anybody waste their time on such unfamiliar system that is not particularly compatible with the miracles and the horrors of the past?

So, I think that they fundamentally rejected the new system.

Still, it ultimately broke through anyway, because some people must have found niche applications in which it was incredibly useful. So, niche after niche started converting. Familiarity grew. After a while, the new system was so widespread that the original objections no longer made sense. The last standing irredenti ("We will never surrender!") eventually caved in, and grudgingly adopted it too.

I think that they still couldn't care about number zero in calculations (What for anyway?) after converting to the decimal place system. Does accounting really need it? Does astronomy really need it? (astronomy: navigation by ship on the high seas to move the expensive spices around!) Does anything else really need it? It was just people who philosophized and played with numbers for the sheer sake of doing that, who used zero as a number. In a sense, mathematics is just a game, and the number zero will then eventually emerge out of that game. Intrinsically, however, it does not have a use in itself, outside that game. Of course, when science and engineering started committing to all of that, things became a bit different. From there on, there were real-life implications too. However, it is not mathematics itself that does that. It is the downstream users in empirical, real-world subjects that cause that transformation.
• 4k
In my impression, the systematic use of the digit zero became a necessity with the introduction of the decimal (positional) system. For example, the number 504 has a zero in the middle, because of the mere bureaucratic-administrative formalisms imposed by the decimal place system. It only has a syntactic meaning.

As you might have noticed, ancient zeros appear in the context of calendars or arithmetic with numbers like 504 or 50 (Bakhshali manuscript if I remember correctly) wherein zero is more of a placeholder than a real number.

Of course the Bakhshali zero in 50 was part of a computation process involving arithmetic with zero and I think that's what makes it so valuable. Zero was a number in it's own right.

If we consider basic arithmetic we have four operations:
2. Subtraction
3. Multiplication
4. Division

In terms of real world problems none of the four operations except subtraction requires zero as a number.

Nobody to my knowledge ever frames questions with zero over the operations of addition, multiplication and division. I tried to think of some problems I could give to an elementary student but it's quite impossible (maybe it's just me). However subtraction questions are easy to come up with e.g. if I have 5 dollars and give it all away to my friend, how many dollars do I have left with me?

[ I just realized that addition, being the inverse of subtraction, can be used to frame questions where the answer is zero e.g. How much money did my friend give me if I started off with 5 dollars and end up with 5 dollars?]

I'm guessing here so consider it carefully...I think zero as a number was born in subtraction problems where the minuend and the subtrahend are equal.

Later on, with the development of algebra multiplication by zero cropped up in problems e.g. 5y + 3 = 3 and division followed naturally as it's the inverse of multiplication.

What do you think?
• 702
I'm guessing here so consider it carefully...I think zero as a number was born in subtraction problems where the minuend and the subtrahend are equal.

Yes, I think that accountants have procedures in which the sum in one column must be equal to the sum in the other column.

If these numbers are, for example, not supplied in order, you can still verify this equality by substracting the numbers in the second column instead of adding them. The final result must then be zero.

Still, they did not document much back then, and with many of these scant, historical documents now gone, they have left us guessing. Even of Algorithmi's original book, "The Art of Hindu Reckoning", there is no extant copy left. We only have access to a translation.
• 811
↪fishfry ↪alcontali Thank you. Sorry for being lazy about this but do you have an idea about what kind of problems Brahmagupta was dealing with when he needed to formalize zero/nothing?

Me? Not my bailiwick I'm afraid.
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