Number system

Number System – Advanced Study Notes

Audience: Graduate students, quantitative‑aptitude aspirants, and anyone interested in the mathematical foundation of numeric representations.
Goal: Build a solid, first‑principles understanding of the number system, its algebraic properties, base‑conversions, and modular arithmetic, culminating in a challenging problem that tests both theory and application.

Notation
• ℕ – natural numbers (1,2,…)
• ℤ – integers
• ℚ – rational numbers
• ℝ – real numbers
• ℂ – complex numbers
• ℤₙ – integers modulo n
• b – base (integer b ≥ 2, but we’ll discuss negative and non‑integer bases later)

1. Foundations of the Number System

1.1 Peano Axioms (Natural Numbers)

From these axioms one constructs addition and multiplication recursively:

• Addition
  n + 1 = S(n);
  m + S(n) = S(m + n).

• Multiplication
  n·0 = 0;
  n·S(m) = n·m + n.

These recursive definitions imply associativity, commutativity, distributivity, and the existence of additive/multiplicative identities and inverses (for ℤ).

1.2 Construction of Integers

Define an equivalence relation on ℕ×ℕ by
(a,b) ~ (c,d) ⇔ a + d = b + c.
The equivalence class [a,b] represents the integer a - b. This construction yields:

• Addition: [a,b] + [c,d] = [a+c, b+d].
• Multiplication: [a,b]·[c,d] = [ac + bd, ad + bc].

The integers ℤ are then the set of all such equivalence classes, with 0 = [0,0] and 1 = [1,0].

1.3 Construction of Rationals

Define an equivalence relation on ℤ×ℤ* (where ℤ* = ℤ \ {0}) by

(a,b) ~ (c,d) ⇔ ad = bc.
The equivalence class [a,b] represents the rational number a/b.

• Addition: [a,b] + [c,d] = [ad + bc, bd].
• Multiplication: [a,b] · [c,d] = [ac, bd].

The rationals ℚ form a field: every non‑zero element has a multiplicative inverse.

1.4 Real Numbers (Dedekind Cuts / Cauchy Sequences)

A Dedekind cut is a partition (L,R) of ℚ such that:

1. L and R are non‑empty, disjoint, L∪R = ℚ.
2. Every element of L is less than every element of R.
3. L contains no greatest element.

Each cut represents a real number. If L has a greatest element, the cut corresponds to a rational; otherwise it represents an irrational.

(We will not delve into the full construction here, but it is essential to know that ℝ is the completion of ℚ with respect to the metric d(a,b)=|a-b|.)

1.5 Complex Numbers

Define ℂ = ℝ × ℝ with addition (a,b)+(c,d)=(a+c,b+d) and multiplication (a,b)·(c,d)=(ac-bd, ad+bc).
Identify i = (0,1); then ℂ is the field of numbers a+bi with i² = -1.

2. Algebraic Properties of the Number System