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 $b \neq 0$, 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 — making $(\mathbb{Z}, +)$ an abelian group rather than merely a monoid. Rationals arise by allowing division by nonzero numbers, making $\mathbb{Q}$ a field. 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.