Why do some accept zero as a natural number but others don't?

Question

I have had many teachers who have told me that zero is a natural number but then there is those teachers who say its not. why is that ?

Answer 1

The natural numbers are those we use for counting objects. Some authors include $0$ as a 'natural' number of objects to have and some don't. There is no consensus in the mathematical community. We do know that the number zero came much later in human history than the rest of them, so some argue that it is less 'natural', but on the other hand it is perfectly normal to consider zero objects (not as abstract as considering -4 objects) so it may be a natural number. It depends on the author.

Answer 2

I think this is primarily based on peoples' differing definitions, not any fundamental misunderstanding. There's no harm done when a student is learning grade school math.

However, using a "typical" construction of the natural numbers using set theory, $0$ is a natural number. The natural numbers are defined by using a function

$$ S(a) := a \cup \{a\}$$

so that $n+1 := S(n)$, and $0$ is defined as the empty set. Starting at $0$ and not at $1$ is then consistent with the definition of the size of a set (cardinality): two sets have the same size if there exists a bijection between them. Since the number $1$ is defined by

$$1 := \varnothing \cup \{\varnothing\} = \{ \varnothing\}$$

then it has one element, as you might expect.

Answer 3

I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it.

Pros of considering $0$ not to be a natural number:

• generally speaking $0$ is not natural at all. It is special in so many respects;

• people naturally start counting from $1$;

• the harmonic sequence $1/n$ is defined for any natural number n;

• the $1$st number is $1$;

• in making limits, $0$ plays a role which is symmetric to $\infty$, and the latter is not a natural number.

Pros of considering $0$ a natural number:

• the starting point for set theory is the emptyset, which can be used to represent $0$ in the construction of natural numbers; the number $n$ can be identified as the set of the first $n$ natural numbers;

• computers start counting by $0$;

• the rests in the integer division by a $n$ are $n$ different numbers starting from $0$ to $n-1$;

• it is easier to exclude one defined element if we need naturals without zero; instead it is complicated to define a new element if we don't already have it;

• integer, real and complex numbers include zero which seems much more important than $1$ in those sets (those sets are symmetric with respect to $0$);

• there is a notion to define sets without $0$ (for example $\mathbb R_0$ or $\mathbb R_*$), or positive numbers ($\mathbb R_+$) but not a clear notion to define a set plus $0$;

• the degree of a polynomial can be zero, as can be the order of a derivative;

I have seen children measure things with a ruler by aligning the $1$ mark instead of the $0$ mark. It is difficult to explain them why you have to start from $0$ when they are used to start counting from $1$. The marks in the rule identify the end of the centimeters, not the start, since the first centimeter goes from 0 to 1.

An example where counting from $1$ leads to somewhat wrong names is in the names of intervals between musical notes: the interval between C and F is called a fourth, because there are four notes: C, D, E, F. However the distance between C and F is actually three tones. This has the ugly consequence that a fifth above a fourth (4+3) is an octave (7) not a nineth! On the other hand if you put your first finger on the C note of a piano your fourth finger goes to the F note.

I would say that in the natural language the correspondence between cardinal numbers and ordinal numbers is off by one, thus distinguishing two sets of natural numbers, one starting from 0 and one starting from 1st. The 1st of January was day number $0$ of the new year. And zeroth has no meaning in the natural language...