# Is every natural number positive?

Jan 19, 2017

Yes.

#### Explanation:

Natural numbers, also called counting numbers, are numbers you would normally count with. A good rule of thumb is that if you could count it on your fingers it's a natural number (double digits, triple digits are included too though). Negative numbers CANNOT be considered natural numbers.

Jan 19, 2017

It depends on if you consider 0 a natural number. If it is then every natural number is non-negative.

#### Explanation:

It depends on whether or not $0 \in \mathbb{N}$. If not, then yes every natural number is positive. If 0 is a natural number then every natural number is non-negative. Some people insist that $0 \notin \mathbb{N}$, and others say the opposite.

Jan 19, 2017

It depends...

#### Explanation:

Natural numbers are always non-negative and whole, but sometimes $0$ is considered a natural number and sometimes it is not.

One way of thinking of it is as follows:

A cardinal number is a number that counts the number of items in a set. Since it counts items, it is whole and cannot be negative, but the empty set is a set containing $0$ elements. So $0$ is a cardinal number.

An ordinal number tells you the order in which items occur. Ordinal names are "first", "second", "third", etc., corresponding to numbers $1 , 2 , 3 , \ldots$. It is natural to start from $1$ when speaking of a sequence of items. Like cardinal numbers, ordinal numbers are whole and non-negative, but $0$ is not an ordinal number.

Natural "counting" numbers could refer to a cardinal or ordinal type of usage. So there are good reasons for starting from $0$ or from $1$. It really depends on the context. Since the symbol $\mathbb{N}$ for the natural numbers is ambiguous, it may be best to instead use ${\mathbb{N}}_{0}$ to stand for natural numbers starting from $0$ and ${\mathbb{N}}_{1}$ for natural numbers starting from $1$.