What Are Rational Numbers?

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where p is the numerator and q is the denominator, with the condition that q ≠ 0.

Rational numbers include all integers, fractions, and terminating or repeating decimals. They form a dense set, meaning between any two rational numbers, there exists another rational number.

Examples of Rational Numbers:

Integers: 5 (which can be written as 5/1), -3 (as -3/1), 0 (as 0/1)

Fractions: 1/2, 3/4, -7/8

Terminating decimals: 0.75 (which is 3/4), 2.5 (which is 5/2)

Repeating decimals: 0.333... (which is 1/3), 0.1666... (which is 1/6)

Visualizing Rational Numbers

Rational numbers can be visualized on a number line. Each rational number corresponds to a unique point on the line, though interestingly, there are points on the number line that don't correspond to rational numbers (these are irrational numbers).

1/4
1/2
1
3/2
7/4

Fraction Visualization

Fractions represent parts of a whole. The denominator shows how many equal parts the whole is divided into, and the numerator shows how many of those parts we're considering.

1/2

One half

2/3

Two thirds

3/4

Three quarters

Properties of Rational Numbers

Property Description Example
Closure The sum, difference, or product of any two rational numbers is also a rational number. 1/2 + 1/3 = 5/6
Commutativity Order doesn't matter in addition or multiplication. 1/2 + 1/3 = 1/3 + 1/2
Associativity Grouping doesn't matter in addition or multiplication. (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4)
Distributivity Multiplication distributes over addition. 1/2 × (1/3 + 1/4) = (1/2 × 1/3) + (1/2 × 1/4)
Identity Elements 0 is the additive identity, 1 is the multiplicative identity. 1/2 + 0 = 1/2, 1/2 × 1 = 1/2
Inverse Elements Every rational has an additive inverse, every nonzero rational has a multiplicative inverse. Additive inverse of 1/2 is -1/2, multiplicative inverse is 2/1

Operations with Rational Numbers

Try It Yourself: Add Two Fractions

Enter two fractions to see their sum:

/ + /

How to Perform Operations:

Addition/Subtraction: Find a common denominator, then add/subtract numerators.

Multiplication: Multiply numerators together and denominators together.

Division: Multiply by the reciprocal of the divisor.

Rational vs. Irrational Numbers

While rational numbers can be expressed as fractions of integers, irrational numbers cannot. Irrational numbers have decimal expansions that neither terminate nor become periodic.

Examples of Irrational Numbers:

√2, π (pi), e (Euler's number), φ (the golden ratio)

These numbers cannot be expressed as simple fractions and their decimal representations go on forever without repeating.