# Is 0 a rational, irrational, natural, whole, integer or real number?

Jun 28, 2015

$0$ is a rational, whole, integer and real number.

Some definitions include it as a natural number and some don't (starting at $1$ instead).

#### Explanation:

Natural numbers are the numbers $1 , 2 , 3 , \ldots$ or the numbers $0 , 1 , 2 , 3 , \ldots$ according to whose definition you choose. So $0$ may be considered a natural number or not.

Whole numbers are the numbers $0 , 1 , 2 , 3 , \ldots$

Integers include negative numbers, but not fractions. So the integers are: $0 , 1 , - 1 , 2 , - 2 , 3 , - 3 , \ldots$

Rational numbers are any number that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. So $5$, $12.42$, $- \frac{17}{3}$ and $0$ are rational numbers.

There are infinitely many rational numbers, but they do not form a continuous line. The continuous line of numbers is called the real number line. It includes all the previous numbers we have mentioned, but also numbers like $\sqrt{2}$, $\pi$ and $e$, which are not rational. Some real numbers - such as $\sqrt{2}$ - are the roots of polynomials with integer coefficients. These are known as algebraic numbers.

Irrational numbers are any real numbers that are not rational. So $0$ is not an irrational number.

Some (in fact most) irrational numbers are not algebraic, that is they are not the roots of polynomials with integer coefficients. These numbers are called transcendental numbers. $\pi$ and $e$ are both transcendental numbers.