Lesson 3: Integers and Rationals

Entry conditions

Extend $\mathbb{N}$ only when you can construct larger number systems from explicit equivalence relations or pairs.

Definitions

• Integers $\mathbb{Z}$ can be constructed as equivalence classes of pairs $(a,b)$ of natural numbers, with $(a,b)\sim (c,d)$ if $a+d=b+c$.
• Rationals $\mathbb{Q}$ can be constructed as equivalence classes of pairs $(a,b)$ with $b\ne 0$, with $(a,b)\sim (c,d)$ if $ad=bc$.

Vocabulary (plain language)

• Equivalence relation: a relation that is reflexive, symmetric, and transitive.
• Equivalence class: a set of elements considered the same under the relation.

Symbols used

• $\mathbb{Z}$: integers.
• $\mathbb{Q}$: rationals.

Intuition

Integers arise by allowing subtraction to be always possible. Rationals arise by allowing division by nonzero numbers. Both are built explicitly from natural numbers, not assumed.

Worked example

The integer $-1$ can be represented as the equivalence class of $(0,1)$, since it behaves like “subtract 1.” The rational $1/2$ can be represented by the class of $(1,2)$.

How to recognize the structure

• You can define the equivalence relation clearly.
• You can show arithmetic operations are well-defined on classes.

Common mistakes

• Treating negative numbers as primitive without a construction.
• Using fractions without specifying a nonzero denominator.