The History and Concept of the Number 0
The goal of this FAQ is to clear up the concept of 0 and specifically the operations that are allowed with 0.
The best way to start this FAQ is to look at a bit of history
Table of Contents
A short history of 0
Historically, there are two different uses of zero: zero as a placeholder and zero as a number in its own right. Zero is used as a placeholder in numbers such as 1010. It merely indicates that the first one stands for a thousand and that the second 1 stands for ten. If we didn’t have a placeholder zero, then we couldn’t differentiate between 11 and 1010. So zero is just a symbol here, it doesn’t have any particular meaning. Zero as a number is a much more controversial topic. It asserts that zero is a number in its own right and has the same privileges as 1,2,3,4, etc. Nowadays, zero is a well-respected number, but that wasn’t always the case. It didn’t even always exist as a placeholder. For example, the Babylonians couldn’t differentiate between numbers such as 8 and 80. Nevertheless, people soon saw the difficulty and started using a placeholder anyway. Our concept of “zero” as both a placeholder and a number originated in India in the 9th century. However mathematicians were quite unsure about how to work with zero. For example, some rules involving zero were
A number when divided by 0 is a fraction with 0 in the denominator.
Zero divided by zero is zero.
As we shall soon see, these rules are not in practice today.
The Indian mathematical knowledge soon found its way to the Arabs. Specifically, al-Khwarizmi popularized the Hindu numeral. The Arab knowledge found its way into Europe by Fibonacci. By the 16th century, the Hindu numerals (including zero) were in use throughout Europe.
For more information about this fascinating topic, see: http://www.gap-system.org/~history/HistTopics/Zero.html
Is zero positive or negative?
By very definition, zero is defined to be neither. The positive numbers are defined as the numbers x such that x>0. The negative numbers are defined as the numbers x such that x<0. So 0 is neither positive nor negative. We also say that 0 has no sign. This is true by definition, so that might make it pretty arbitrary: that is, other definitions can be used but they are not standard.
Zero is also both nonnegative and nonpositive.
Operations with zero
Zero is known as the additive identity. That is: for every number x (that is: x can be natural, real, complex, etc.) we have that
From this, we can easily infer what 0x is (by using distributivity)
So 0 times anything is 0. (This holds in every ring.)
The division is a bit more difficult. First, we must define what division is. We say that
From this, we can infer that
Division by 0
Now, what if 0 is the denominator? That is, what if we have
This argument fails when x=0. In that case, there are numbers c such that 0c=0. The problem is that these numbers are not unique. All numbers satisfy 0c=0. So
Some say that
Of course, it is possible to extend our number system to include
We see that the graph of
But what if we would have a strange number system where
Exponentiation and 0
Exponentiation and 0 is another messy topic. Recall that if n is a nonzero natural number, then we define
Well, to formulate a sensible definition, we first need to know something more about exponentiation. That is, we should know about the formula
So we see that
What is the problem with
However, many mathematicians disagree with this and do define
But if x=0 and n=0, then this sum becomes undefined because we have to evaluate
Factorials and 0
We have seen that it’s not possible to define
Firstly, the factorial can be defined recursively as
So, a priori, we cannot know what 0! is. But what if we apply the recursive equation on n=0, then we get
So, we see that defining 0!=1 would make sense.
Of course, the factorial has a very practical meaning. Indeed, n! is the number of ways we can order n objects. For example, there are 3!=6 possible ways to order 3 elements. Indeed, we can order {a,b,c} as
There are 2!=2 ways to order 2 elements. Indeed, we can order
There is 1!=1 way to order 1 element. Indeed, we can order
And finally, there is 0!=1 way to order 0 elements. Indeed, we can order
The factorial n! also gives the number of bijections
Another way to use factorials is when looking at binomial coefficients. We define
This gives us the number of ways to choose k elements from n element, where order doesn’t matter. For example
is the number of ways to choose 2 elements from a set of 4 elements. Indeed, from the set
What happens in our formula if k=n?? Then we should get the number of ways to choose n elements from a set of n elements. There should only be one way of doing it. Thus
What if k=0?? Then we should get the number of ways to choose 0 elements from n elements. There should also be 1 way of doing this: selecting no element. And indeed:
Also, our binomial theorem only works if we define 0!=1.
Lastly, the final result is given by the Gamma function. The Gamma function is a continuous function that is defined as
This looks strange, but the Gamma function has the property that
So the Gamma function can be seen as a continuous continuation of the factorial. Now, we have
as can be easily calculated or which can be seen on this graph:
All this evidence points to the inevitable conclusion that we have to define 0!=1. There is absolutely no problem with this definition, as opposed to defining
The following forum members have contributed to this FAQ:
Borek
D H
HallsOfIvy
Hootenanny
Mark44
micromass
tiny-tim
Advanced education and experience with mathematics
Good article!
“Just a historical correction. Khwarizmi was actually Iranian.”
Actually, micromass never said he was other, and as a Muslim, it would have been through the Arabs and with Arabs that the information came.
Just a historical correction. Khwarizmi was actually Iranian.
Micromass, you omit to mention that the Mayas also created or came up with the number Zero. :D Good analysis.
Nice article !
Thanks, Micromass.
Nice article, Micro! Thanks for sharing!