Rational numbers

Builds on

Required prerequisites, all the way down: 6

You will learn to

• Add, subtract, multiply and divide rational numbers, including negative ones.
• Explain why every non-zero rational number has a reciprocal, in contrast with the integers.
• Convert between fractions and decimals, and show that a fraction's decimal expansion terminates or eventually repeats.

Definition

• A rational number is a number a/b with a and b integers and b ≠ 0; a/b = c/d when ad = bc.
• The reciprocal of a non-zero rational number a/b is b/a.

Theorem

• ℚ is closed under addition, subtraction, multiplication, and division by non-zero numbers.
• The decimal expansion of a rational number terminates or eventually repeats, and every such decimal represents a rational number.
• Between any two distinct rational numbers there is another, for example their average.

Common misconceptions

• Thinking division always makes smaller; dividing by 1/2 doubles.
• Believing 0.999… is less than 1.
• Thinking the number line is full once every fraction is placed; √2 is still missing.

Leads to

Optional discoveries

• Algebraic structures: monoids, groups and rings ℚ is a field: every non-zero element has a multiplicative inverse.
• Countability and Cantor's diagonal argument Although the rationals are dense, they can be listed one after another: ℚ is countable.

Learning pathways

• Conventional K–12 progression
• Adult beginner pathway