Rational Number

What is a Rational Number?

A rational number is a number that can be expressed as the quotient or fraction $\frac{a}{b}$ of two integers, where $a$ is the numerator and $b$ is the denominator ( $b\ne0$ ). In other words, a rational number is any number that can be represented as a fraction of two integers.

Fractional Form

Rational numbers can be written in the for $\frac{a}{b}$ , where $a$ and $b$ are integers and $b\ne0$ .

Repeating or Terminating Decimal

The decimal representation of a rational number either terminates (ends) or repeats (has a repeating pattern) after a certain point.

Integer as a Special Case

Every integer is a rational number because it can be written in the form $\frac{a}{1}$ .

Example of Rational Numbers

Whole Numbers:
- $-3$ : This integer is a rational number because it can be written as $\frac{-3}{1}$ .
Fractions:
- $\frac{2}{3}$ : This is a rational number because it is the quotient of two integers, where the numerator is $2$ and the denominator is $3$ .
Decimals with a Repeating Pattern:
- $.333\dots$ : This decimal representation is rational because it has a repeating pattern (denoted by the ellipsis).
Decimals with a Terminating Pattern:
- $0.25$ : This decimal representation is rational because it terminates after a finite number of digits.
Negative Rational Numbers:
- $-\frac{4}{5}$ : This negative fraction is a rational number because it is the quotient of two integers.
Mixed Numbers:
- $2\frac{1}{4}$ : This mixed number is rational because it can be converted to the fractional form $\frac{9}{4}$ .

Not Rational Numbers

Irrational Numbers:
- $\sqrt{2}$ : The square root of $2$ is not a rational number because it cannot be expressed as the quotient of two integers, and its decimal representation is non-repeating and non-terminating.
Non-Repeating, Non-Terminating Decimals:
- $\pi$ : The mathematical constant pi ( $pi$ ) is not a rational number because its decimal representation goes on forever without repeating.
Variables:
- $\frac{x}{y}$ : This algebraic expression is not a specific rational number unless values are assigned to $x$ and $y$ .