# What's the hard part of zero?

A lot of textbooks said it was hard for human to accept zero when it was first introduced.

How could it be? It seems to me as natural as positive integer which represent there is no elements at all.

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Related note: it's not just zero that's hard. All numbers were originally hard. There are human groups that have no number words, and no concept of quantification more specific than "few" and "much". Naturally, these groups have a difficult time learning to count. We don't think about this much because numeration was invented in prehistory, and most human groups have practiced it for so long that there is no record of the transition. Nonetheless, the existence of isolated pre-numerate tribes attests to the fact that number had to be invented at some point. –  JSBձոգչ Nov 30 '10 at 20:39
See also www-groups.dcs.st-and.ac.uk/~history/HistTopics/Zero.html BTW, Euclid also did not consider $1$ to be a number. See aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html –  Robert Israel Dec 25 '11 at 9:10

The problem was that math started out of necessity. Shepherds needed to count how many sheep they own, parents needed to count how many children they had, etc. For most simple mathematics, zero is simply a place-holder. Who needs to count zero sheep? Even the most basic forms of counting--using your fingers--who has zero fingers? Zero is also pretty closely tied to the concept of negative numbers as well, which are a very non-natural inclusion for mathematics that simply evolve out of necessity. What shepherd in his right mind has -1 sheep. Wrapping your head around the idea of zero is easy once the idea is conveyed, but somewhat non-intuitive to arbitrarily include. Even worse, once you do include zero, how do you deal with all the problems that come along with it such as division by zero. A great history on the story of how we came to understand zero can be found in the book: Zero: The Biography of a Dangerous Idea by Charles Seife.

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I think "necessity" is too strong of I word, I'd fromulate it as "need". –  NikolajK May 16 '13 at 9:28

Of course it seems natural to you; you grew up in the modern world, where everyone accepts zero. More importantly, people now accept the abstract concept of numbers and are capable of divorcing them from the things that they represent. This is a sophisticated point of view. From a more naive point of view, a number is a property of a collection of objects: when I say there are $2$ apples somewhere, that is a property of a collection of apples. If there are no apples, then what is there to "hold" the corresponding property? So instead of saying there are $0$ apples, people said there are no apples.

In computer science terms, apples.num() is not defined if there is no apples variable!

You also have to understand that "human" in your statement means (from my understanding) "Western mathematicians." Indian mathematicians had no trouble with zero.

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+1 Indian mathematicians had no trouble with Zero –  Teja Kantamneni Nov 29 '10 at 12:09
Yep, in a way, it has been a 'problem' within Western mathematics because we inherited it from the Greeks, who were fantastic geometers. But as a consequence, if a number could not be represented by a geometric construction, they were struggling with how to place it. –  Raskolnikov Nov 29 '10 at 14:46
The OP says that humans had trouble with zero when it was first introduced. Zero was introduced much earlier in Indian mathematics than in Greek/Western mathematics. But presumably there was still an identifiable point in history when Indian mathematics did not have the concept of zero. Or are you saying that there is something inherent in (ancient) Indian language/culture that makes the concept of zero more natural? –  Pete L. Clark Nov 29 '10 at 16:34
@Pete: I'm not willing to make quite so strong a claim without consulting sources, but my recollection of what I've read in popular accounts suggests that something like that may have been true. –  Qiaochu Yuan Nov 29 '10 at 17:00
Well, it would be interesting to hear about that. If you can find a reference, let me know. –  Pete L. Clark Nov 29 '10 at 22:38

Question: What is the name of the first female U.S. president?

Answer: There hasn't been one yet.

It is common to respond to a question by explaining that the question itself is flawed. Because there have been no female U.S. presidents yet, it doesn't make sense to ask for the name of the first one.

Question: How many apples do you have?

Answer: I don't have any apples.

Responding to a "how many" question can follow the same principle: the question is flawed because I don't have any apples, so it doesn't make sense to ask how many I have. It can be very, very difficult to revise one's thinking about something so basic; even today, there are many people who can't understand zero as a number, and instead think of it as simply a computational trick, interpret it as a sort of "not a number", or other similar sort of thing.

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You can't count out zero 'things'; I would imagine that that was the big issue. Same goes for the negative side of $\mathbb{Z}$.

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Let's see: how many times have I been to Prague? Zero. Didn't I just...? –  Pete L. Clark Nov 29 '10 at 22:39
@Pete, nice one :) but I think in some cases it might be more natural to apply the concept of "Null" to similar questions rather than "Zero". How many times have I been to Prague? Well, I haven't traveled outside of my country. How many sheep do I have? I'm not a shepherd, therefore I don't have sheep. What's the average number of sheep that people own? Well, 95% of the population are not shepherds, so should they count as having zero sheep, or should they be excluded from the calculation? I'm just saying that sometimes it makes sense to use zero and sometimes it doesn't. –  Dr. Wily's Apprentice Nov 30 '10 at 20:08
@Dr. Wily's Apprentice: Yes, perhaps the best definition of zero is as the cardinality of the null set, so there's a definite relation there. Perhaps interestingly, zero seems easier to understand than the null set: philosophically speaking, nothingness is a profound and difficult concept, whereas zero is "merely" a formal symbol used in the accounting of nothing. –  Pete L. Clark Nov 30 '10 at 20:26
To address to your point as to whether one should count zero sheep or exclude them entirely: yes, at some level it's a problem, but not at the level of accounting. The whole point of zero is that you can choose to add it in to your calculation or not: either way leads to the same final tally. –  Pete L. Clark Nov 30 '10 at 20:29
@Pete: interesting point regarding the philosophy of nothingness. Just to clarify my "Average number of sheep" example; my intention was to point out that including the relatively large number of non-shepherds in a population with a value of zero sheep when calculating will lead to a much smaller result as compared to only calculating the average number of sheep among people who actually are shepherds. i.e. let's say all shepherds have 20 sheep; everyone else does not have sheep. If there are 100 people but only 5 shepherds, then the desired average might be 20, or it might be 1. –  Dr. Wily's Apprentice Nov 30 '10 at 20:54

There are actually five different kinds of zero. Just because some parts of zero are not implemented, it does not mean that any have. Egypt did have a zero, but only in the sense of 'empty bag'.

If you have stones in columns of your abacus, the number system allows you to convert these into some portable form, so someone else can repeat the pattern. In token systems, one gives the name of a token for each stone. Positions with no tokens need no name. Here is the roman layout for such a thing. A number like 56, is an L, a V and an I stone, ie LVI.

      D     L     V
-------------------      The inca used a similar based on 1 and 5 in
CC    XX    II         each column of 20.
CC    XX    II


The demotic system, repeated into alphabetic systems, use a system where there is a seperate symbol for 'seven stones in the tens-column'. Like the tokenised system, there is no need to present an empty column.

You only need empty column-markers, when the symbols consist purely of 'six stones in this column'. The usual implementations are to skip outside columns, and skip internal columns.

For example, the greekish system (borrowed by the arabs etc), is to use the '10' letter as a kind of zero (which is why it appears at the end of the row of numbers).

Leading zeros appear in division-bases, like our time system (division-base 60), where in 0001 it means 0 in the first column and 1 in the second column. In base 60, there is an alternation between the top and bottom half of the abacus (which is presented as horizontal vs vertical arrows). Zeros were represented by points (ie sentence stops).

     h    m    s
0    0    0        00:01:00 and 00:01:  mean the same thing!
0    1    0


Trailing zeros appear in multiple-bases, where the units are the lowest column, so we have

     m    c    x    i
4    0    3    0


Both the 0's at c and i are needed, the first says an empty column lies between m and x, and the second says the last quoted digit is an x, rather than an i.

Semimedial zeros do not occur in decimal, but you can readily find these in base 60. The actual implemetation of the sumerian digits is not eg '35' as a digit, but two different runs of digits, eg

                  Sumerian                                Mayan

C2                 C   2
A-F         C = 30          C = 30      -             A-C  (5,10,15)
----------              vs                            ----
1-9         2  = 2             -       2 = 2          0-4

32                30  :   02


Both of these would be written C (three horizontal rays) followed by 2, two verticals, but we would write these as 32 vs 30.02 The usual is to space the C from the 2, to show that there are empty places, ie C 2 vs C2.

The Mayans never had this problem, for the reason that the lower position 0-4 is always written where the column is significant. It's kind of like writing base 30 numbers as 0 to 9, with an optional ' and " for +10 and +20 respectively: eg 5 = 5, 5' = 15, 5" = 25.

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When did 'none' come to be a number? My guess is that it was when merchants or scribes first codified symbolic addition and subtraction to calculate quantities of food and supplies on hand. Intuitively, they must have known that 'none' would be the result of subtracting 100 bushels of wheat from same. And that adding 100 bushels to 'none' would give you a measurable quantity. Philosophers may have quibbled about whether 'none' was a real and true number, but I'm sure it was all too real to the merchants and scribes of antiquity.

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In many places, i.e., accounting 0 meaning nothing is represented by a line.

for example $3.— for exactly 3 dollars. rather than$3.00

and sometimes by xx ... as written on a cheque:

---- three dollars ----- xx/100

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Interesting question (which I edited out): how does one represent an em-dash? In standard TeX and LaTeX you may know that an em-dash is typed using three adjacent hyphens, ---. However, that won't work in our particular LaTeX/HTML environment. Take a look at the source for my edit for a clunky, nonstandard, unattractive workaround. –  Rick Decker Jul 5 '12 at 20:25
@Rick, David: You can find the em dash character in a character map application, or just copy it from the Wikipedia page like I just did. That said, I don't think this post answers the question. –  Rahul Jul 5 '12 at 20:29
@Rahul Clever. I hadn't thought of a simple copy/paste solution. That said, I agree that the post fails to answer the question, though the question itself is a good one. –  Rick Decker Jul 6 '12 at 2:09
The example shows a valid use of zero, in the sense of an empty column. Here the column has a unit, and the dash prevents the column being filled. –  wendy.krieger May 16 '13 at 10:53