Number Theory Domains

Number theory primarily focuses on integers but utilizes many number sets as essential tools. Explore how each number set contributes to this fascinating mathematical field.

Natural Numbers (ℕ)

The foundation of number theory. These are the counting numbers starting from 1 (or sometimes 0).

Number theory studies properties like primeness, divisibility, and distribution of these numbers.

Examples: 1, 2, 3, 4, 5, ...

Integers (ℤ)

Includes all natural numbers plus their negatives and zero. The primary domain of classical number theory.

Key concepts include Diophantine equations, modular arithmetic, and the distribution of primes.

Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational Numbers (ℚ)

Numbers that can be expressed as fractions of integers. These are directly involved in many number theory problems.

Finding rational solutions to equations is a fundamental part of the field.

Examples: 1/2, -3/4, 2, 0.75, -5

Irrational Numbers

Numbers that cannot be expressed as fractions. They often appear as results in number theory.

Proofs of irrationality (like for √2 or π) are classic number theory problems.

Examples: √2, π, e, φ (golden ratio)

Complex Numbers (ℂ)

Numbers with real and imaginary parts. These are powerful tools in analytic number theory.

The Riemann zeta function, defined for complex numbers, provides deep insights into prime distribution.

Examples: 3+4i, -2i, 5 (real part), i (imaginary unit)

Number Sets Visualization

The number line shows examples of different number types and their relationships:

Note: Complex numbers cannot be represented on a simple number line as they have two components (real and imaginary).

Number Theory Domains | Understanding the different number sets in mathematics