Number System – Advanced Quantitative Aptitude Notes
For USMLE Quantitative Aptitude, but written at MIT‑level rigor
1. Introduction
The “number system” is the mathematical framework that underlies every quantitative problem: it tells us what numbers exist, how they are built from one another, and how they are represented in different bases.
In the USMLE context we usually encounter:
| Type | Typical appearance | Example |
|---|---|---|
| Natural numbers | Counting, age, serial numbers | 1, 2, 3 … |
| Integers | Dosage adjustments, temperature changes | –5, 0, 12 |
| Rationals | Lab values (e.g., 0.75 g/dL), ratios | 3/4 |
| Reals | Continuous measurements | 3.14159 |
| Complex | Rarely in USMLE, but good to know | 5 + 2i |
| Base‑b representations | Digital data, binary computers | 1011₂, 7F₁₆ |
We will start from first principles, build each set formally, prove representation theorems, discuss edge cases, and finish with a challenging problem that requires deep understanding of the concepts.
2. Foundational Axioms (Peano & Set‑Theoretic Constructions)
| Concept | Formal foundation |
|---|---|
| Natural numbers | Peano axioms: 0 ∈ ℕ, ∀n∈ℕ, S(n)∈ℕ, no two distinct numbers have the same successor, and induction holds. |
| Integers | Constructed as equivalence classes of ordered pairs (a,b) ∈ ℕ×ℕ, with (a,b) ~ (c,d) iff a+d = b+c. |
| Rationals | Equivalence classes of ordered pairs (p,q) ∈ ℤ×ℤ{0} with (p,q) ~ (r,s) iff ps = qr. |
| Reals | Dedekind cuts or Cauchy‑complete sequences of rationals. |
| Complex | Ordered pairs (a,b) ∈ ℝ² with addition (a,b)+(c,d)=(a+c,b+d) and multiplication (a,b)(c,d)=(ac−bd,ad+bc). |
| Base‑b representation | For integer (n≥0) and base (b≥2), there exists a unique sequence (a_k…a_1a_0) with (a_i∈{0,…,b−1}) s.t. (n = \sum_{i=0}^k a_i b^i). |
Proof Sketch (Base‑b uniqueness)
Write (n = q_0 b + r_0) with (0≤r_0<b). Then (q_0 = q_1 b + r_1), …, until (q_k=0). The digits are