Rational Number
ArithmeticA rational number is any number that can be expressed as a fraction p/q where p and q are integers and q is not zero.
Formula
\frac{p}{q}, \text{ where } p, q \text{ are integers and } q \neq 0
Definition
A rational number is any number you can write as a fraction (one whole number divided by another, not zero). This includes regular fractions, whole numbers, and decimals that end or repeat.
Example
$3/4$, $-2$, $0.5$, and $0.333\ldots$ are all rational. $3/4$ is already a fraction. $-2 = -2/1$. $0.5 = 1/2$. $0.333\ldots = 1/3$.
Key Insight
The word "rational" comes from "ratio." If you can write the number as a ratio of two whole numbers, it is rational.
Definition
A rational number is any real number expressible as $p/q$ where $p$ and $q$ are integers and $q \neq 0$. Rational numbers include all integers ($n = n/1$), all terminating decimals, and all repeating decimals. A decimal is rational if and only if it terminates or eventually repeats.
Example
$0.75 = 3/4$ (terminates, rational). $0.142857142857\ldots = 1/7$ (repeats, rational). $\pi = 3.14159\ldots$ never terminates or repeats, so $\pi$ is irrational.
Key Insight
Between any two rational numbers there is another rational number (the average of the two). This property is called density. Despite this, most real numbers are irrational.
Definition
The rationals $\mathbb{Q}$ form the unique field of fractions of the integers $\mathbb{Z}$. $\mathbb{Q}$ is a dense, totally ordered field but is not complete: Cauchy sequences of rationals need not converge in $\mathbb{Q}$. The real numbers $\mathbb{R}$ are constructed as the Cauchy completion (or Dedekind completion) of $\mathbb{Q}$.
Example
The sequence $1, 1.4, 1.41, 1.414, 1.4142, \ldots$ is a Cauchy sequence in $\mathbb{Q}$ whose limit is $\sqrt{2}$, which does not exist in $\mathbb{Q}$, demonstrating $\mathbb{Q}$'s incompleteness.
Key Insight
$\mathbb{Q}$ is countable (there is a bijection between $\mathbb{Q}$ and $\mathbb{N}$), yet $\mathbb{Q}$ is dense in $\mathbb{R}$. This means the rationals, though countable, leave no "gaps" that can be detected from within $\mathbb{Q}$ alone.