# Is 1/3 a rational, irrational number, natural, whole or integer?

Jun 26, 2015

$\frac{1}{3}$ is a rational number, being a number of the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$.

It is not a natural number, whole number or integer.

#### Explanation:

Numbers can be classified as follows:

Natural numbers are the numbers $0 , 1 , 2 , 3 , \ldots$ or $1 , 2 , 3 , \ldots$
Some people prefer to start at $0$ and others at $1$.

Whole numbers are the numbers $0 , 1 , 2 , 3 , \ldots$
this is almost the same definition as natural numbers, but does explicitly include $0$.

Integers include negative numbers along with the previous ones, so they are the numbers, $0 , 1 , - 1 , 2 , - 2 , 3 , - 3 , \ldots$

Rational numbers are all numbers of the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. Note that this includes positive and negative integers, since if you let $q = 1$ then $\frac{p}{q} = \frac{p}{1}$ can be any integer.

Real numbers are any numbers on the real line. This includes rational numbers, but also includes numbers like $\sqrt{2}$ and $\pi$, which are not rational.

Irrational numbers are any numbers which are not rational.

Algebraic numbers are numbers which are roots of polynomials with integer coefficients. For example $\sqrt{2}$ is algebraic because it is a root of ${x}^{3} - 2 = 0$. Every rational number is algebraic.

Transcendental numbers are numbers which are not algebraic. They include numbers like $\pi$ and $e$. In fact, most real numbers are transcendental.