Rational Numbers

Rational numbers are ratios of integers and include fractions, integers, terminating decimals, and repeating decimals.

Position on a number line encodes order, sign, and distance. A horizontal number line labels consecutive integers from negative three through positive three. Curved arrows from zero point to highlighted positions on opposite sides, emphasizing magnitude as distance rather than direction.

A rational number is any number that can be written as $a/b$, where $a$ and $b$ are integers and $b\ne0$.

What belongs in the set

Fractions are the obvious rational numbers, but the set is much larger. Every integer is rational because $-7=-7/1$. A terminating decimal such as $0.35$ is rational because $0.35=35/100=7/20$. A repeating decimal is also rational.

The denominator cannot be zero. The expression $4/0$ is undefined, not rational, irrational, or infinite.

Equivalent forms

Multiplying numerator and denominator by the same nonzero number preserves value:

\frac{3}{5}=\frac{6}{10}=\frac{30}{50}=0.6.

Simplest form removes common factors from numerator and denominator. Equivalent forms are useful for different jobs: decimals support measurement, fractions preserve exactness, and percent communicates a comparison to $100$.

Decimal behaviour

A rational number’s decimal expansion terminates or eventually repeats. In lowest terms, a fraction has a terminating decimal exactly when its positive denominator contains no prime factors other than $2$ and $5$.

For example, $7/40$ terminates because $40=2^3\cdot5$. The fraction $2/3$ repeats because its denominator includes the factor $3$.

Turning a repeating decimal into a fraction

The multiplication by $100$ aligns the repeating blocks so subtraction removes the infinite tail.

Common mistakes

Believing rational means positive. Negative fractions and zero are rational.

Treating every long decimal as irrational. A decimal may repeat later; a displayed calculator approximation may also represent an exact rational value.

Adding denominators. $1/2+1/3$ is not $2/5$ because halves and thirds are different-sized parts.

Dividing by zero. A zero numerator is allowed, but a zero denominator is not.

Quick self-check

Can I express the number as a fraction of integers?
Does its decimal terminate or repeat?
Have I kept exact form until a decimal is useful?
Does the final value make sense in its context?