Question

give me every improper fraction

Answer 1

I can't literally list "every" improper fraction because there are infinitely many. I can, however, give the definition, a compact description of the whole set, and ways to generate or enumerate them.

Definition (usual): an improper fraction is a fraction a/b (with integers a, b and b ≠ 0) whose numerator has absolute value at least as large as the denominator — i.e. |a| ≥ |b|. (Some texts restrict to positive denominators and/or positive fractions; the idea is the numerator is not smaller than the denominator.)

Set notation (all improper fractions):
{ a/b ∈ Q : a, b ∈ Z, b ≠ 0, |a| ≥ |b| }.

If you want only positives with positive denominator:
{ n/d ∈ Q : n, d ∈ N, n ≥ d ≥ 1 }.

If you want each rational listed once (reduced form, denominator positive):
{ n/d ∈ Q : n, d ∈ N, n ≥ d ≥ 1, gcd(n,d) = 1 } ∪ the negatives of those.

Properties:
- The set is countably infinite.
- Many improper fractions represent the same rational number (e.g., 4/2 = 2/1); the reduced-form version above gives unique representatives.

How to generate them (simple generator):
- For each denominator d = 1, 2, 3, ...
- For each numerator n = d, d+1, d+2, ...
- output n/d (and -n/d if you want negatives)

If you want them in a single sequence without an infinite inner loop, enumerate by increasing n + d (or by absolute value) and break ties lexicographically; filter to reduced form if you want unique rationals.

Would you like:
- A finite sample (e.g., all improper fractions with denominator ≤ N or numerator ≤ M)?
- Only reduced improper fractions?
- Only positive ones?