Introduction to Number Theory

Definition of Number Theory

• Also known as higher arithmetic
• Study of mathematical interactions and various types of numbers
• Includes odd numbers, even numbers, squares, integers, rational numbers, etc.

Key Focus

• Examines relationships between different types of numbers
• Utilizes formal mathematical proofs to answer questions about these relationships

Historical Example: Euclid and Prime Numbers

• Euclid's inquiry into prime numbers
  • Definition: Numbers divisible only by one and themselves (e.g., 2, 3, 5, 7)
  • Proposed the question: How many primes are there?
  • Hypothesis: Infinite primes exist
• Method of proof:
  • Used proof by contradiction
  • Assumed a finite number of primes and demonstrated the assumption was incorrect

Applications of Number Theory

• Historical view: Considered abstract with no real-world applications
• Modern view: Significant real-world implications
  • Example: Fundamental Theorem of Arithmetic
    • States every number can be uniquely factored into primes
    • Example with the number 12:
      • Factorization: 12 = 2 x 6; 6 = 2 x 3
      • Prime factors are 2, 2, and 3
• Importance in Technology:
  • Used in encryption schemes for business networks
  • Security relies on the unique prime factorization of large numbers

Conclusion

• Number theory is about questioning mathematical concepts and using proofs to explore them
• Practical applications in technology demonstrate its significance in solving real-world problems

---

Call to Action

• Subscribe, like, and check out other Bullis Student Tutors videos.