Rational Number¶

Definition¶

A rational number is a number of the form

Two such fractions represent the same rational number exactly when

where $b,d\neq0\text{.}$ The set of all rational numbers is denoted by $\mathbb{Q}\text{.}$ Math I: Calculus takes $\mathbb{Q}$ and its usual arithmetic as given.

Interpretation¶

Rational numbers represent ratios of integers and enlarge $\mathbb{Z}$ so that division by a nonzero number is possible. Every integer is rational because $a=a/1$$a/1$, so $\mathbb{Z}\subsetneq\mathbb{Q}$$\mathbb{Q}$.

The figure marks the fractions strictly between $0$ and $1$ with denominator at most $9$; the larger dots belong to small denominators such as $\tfrac{1}{2}$, $\tfrac{1}{3}$ and $\tfrac{2}{3}$. As the denominators grow, the dots fill the interval more and more closely. This illustrates the density of $\mathbb{Q}$: between any two distinct rational numbers $p<q$ lies another one, for instance their midpoint $\tfrac{1}{2}(p+q)$$q)$.

The definition separates a number from the way it is written: $\tfrac{1}{2}$, $\tfrac{2}{4}$ and $\tfrac{-3}{-6}$ all represent one rational number, and the test $ad=bc$ decides equality without computing decimals. The condition $b\ne0$ is essential, since no number $x$ satisfies $0\cdot x=1$$1$. Density does not mean that $\mathbb{Q}$ fills the number line: no rational number squares to $2$, so $\sqrt{2}$ is an irrational number; filling such gaps is what the real numbers do.

Key Results¶

• Every rational number has infinitely many representations.

• For $b,d\neq0\text{,}$

  $$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}, \qquad \frac{a}{b}\frac{c}{d}=\frac{ac}{bd},\qquad -\frac{a}{b}=\frac{-a}{b}.$$

• Rational numbers are closed under addition, subtraction, and multiplication, and under division by a nonzero rational number.

• Between any two distinct rational numbers lies another rational number; this is the density of $\mathbb{Q}$ in itself.

Examples¶

• $\dfrac{1}{2}=\dfrac{2}{4}\text{.}$
• $\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}\text{.}$
• $-\dfrac{3}{5}$ is a rational number.
• $\sqrt{2}$ is not a rational number.

Prerequisites¶

• Integer

Related Concepts¶

• Division and Quotient

Sources¶

• Math I: Calculus · Lecture 2

Symbols: element of, integers, rational numbers, square root

Last updated: 22 October 2026 19:04