Number Systems and Arithmetic

The study of mathematics begins with numbers. We build complex systems from simple foundations, starting with the natural numbers.

The Hierarchy of Numbers

1. Natural Numbers ($\mathbb{N}$): $\{0, 1, 2, 3, \dots\}$ (counting numbers).
2. Integers ($\mathbb{Z}$): $\{\dots, -2, -1, 0, 1, 2, \dots\}$.
3. Rational Numbers ($\mathbb{Q}$): Fractions $p/q$ where $p, q \in \mathbb{Z}$ and $q \neq 0$.
4. Real Numbers ($\mathbb{R}$): Includes irrational numbers like $\sqrt{2}$ and $\pi$.
5. Complex Numbers ($\mathbb{C}$): Numbers of the form $a + bi$.

import numpy as np

def classify_number(n):
  if isinstance(n, complex):
      return "Complex"
  if n % 1 == 0:
      return "Integer/Natural"
  return "Fractional/Real"

test_cases = [42, 3.1415, complex(1, 2)]
for n in test_cases:
  print(f"{n} is {classify_number(n)}")

Peano Axioms

The natural numbers are rigorously defined by the Peano axioms, which use a single constant $0$ and a successor function $S(n)$.

• $0$ is a natural number.
• For every $n$, $S(n)$ is a natural number.
• No two numbers have the same successor.
• $0$ is not the successor of any number.

Fundamental Theorem of Arithmetic

Every integer greater than 1 either is a prime number itself or can be represented as a unique product of prime numbers.

def prime_factors(n):
  i = 2
  factors = []
  while i * i <= n:
      if n % i:
          i += 1
      else:
          n //= i
          factors.append(i)
  if n > 1:
      factors.append(n)
  return factors

print(f"Factors of 12: {prime_factors(12)}")
print(f"Factors of 37: {prime_factors(37)}")