Number System — Density of Rational Numbers Between the real numbers 1 and 1000 (exclusive), how many rational numbers exist?

Introduction / Context:
This question tests understanding of the density of rational numbers on the real number line. A rational number is any number that can be expressed as p/q where p and q are integers and q != 0. The concept of density means that between any two distinct real numbers, there are infinitely many rational numbers.

Given Data / Assumptions:

• Interval considered: (1, 1000), open interval (endpoints excluded).
• Rational numbers are of the form p/q with integers p, q and q != 0.
• No restriction on numerator/denominator size beyond being integers.

Concept / Approach:
Rational numbers are dense in the reals: given any two reals a < b, there exists a rational r with a < r < b. Moreover, you can construct infinitely many rationals in any interval by, for example, taking midpoints repeatedly or considering sequences p/n for suitable integers p and large n that land between a and b.

Step-by-Step Solution:
1) Pick any two numbers between 1 and 1000, say 2 and 3; there are infinitely many rationals in (2, 3).2) Because (1, 1000) contains (2, 3), it likewise contains infinitely many rationals.3) More generally, for any n, the set {k/n : k = n+1, ..., 1000n-1} yields many rationals in (1, 1000).4) As n grows, the count grows without bound, proving infinitude.

Verification / Alternative check:
Repeated bisection: start with 1 and 1000; the midpoint 500.5 is rational. Bisect again to get more rationals. This process never ends, establishing an infinite set.

Why Other Options Are Wrong:
998, 999, and 1000 suggest a finite count, which contradicts density. “None” is clearly false because many rationals (e.g., 3/2, 5/3) lie in the interval.

Common Pitfalls:
Confusing “integers” with “rationals” and counting only whole numbers; misreading the question as asking for integers between 1 and 1000.

Final Answer:
infinite