Lecture on 10-Adic and P-Adic Numbers

Introduction to Squaring and Patterns

• Start with the number 5, square it to get 25.
• Square 25 to get 625, and square 625 to get 390,625.
• Pattern: numbers ending in 5 when squared continue the pattern partially as they grow.
• Concept of numbers that are their own square, leading to a system of infinite digits.

Understanding 10-Adic Numbers

• Numbers with infinite digits extending to the left.
• Operations:
  • Addition: Digit by digit from right to left.
  • Multiplication: Last digit depends only on the numbers’ last digits.
• Example: Multiplying a 10-adic number by 7 results in a repeating pattern, showing it equals 1/7.
• Rational numbers can exist without the division symbol.
• Concept of infinite digits similar to decimal repeating patterns like 0.999… equals 1.

Negative Numbers in 10-Adic System

• A 10-adic string of all 9s equals -1.
• Adding 1 to an infinite string of 9s results in 0.
• Subtraction: Adding the negative of a number.

Challenges with 10-Adic Numbers

• The problem arises when a number is its own square.
• Factorization fails due to composite base 10.
• Solution: Use a prime number base (p-adics).

Introduction to P-Adic Numbers

• A prime number base (p-adics) avoids issues seen in base 10.
• Example: 3-adic numbers, with digits only 0, 1, 2.
• Multiplication ensures non-zero results unless a component is zero.

Applications and Importance of P-Adic Numbers

• Used in solving integer-based polynomial equations.
• Essential for modern mathematical problems like Fermat's Last Theorem.

Solving Equations with P-Adics

• Represent equations using powers of a prime (e.g., base 3 for 3-adics).
• Modular arithmetic simplifies finding solutions.
• Example: Solving Diophantus’ sum of squares problem using p-adics.

Geometry and Size in P-Adic Numbers

• Different conception of size and distance, contrasting with real numbers.
• Visualization as a triply branching tree.
• P-adic numbers have a geometric interpretation distinct from real numbers.

Conclusion

• P-adics provide a crucial tool in advanced mathematics, offering solutions not achievable with real numbers.
• Inspiration from historical mathematical discoveries, encouraging exploration and innovation in various fields.

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These notes summarize the lecture's key points on 10-adic and p-adic numbers, focusing on their operations, challenges, applications, and importance in modern mathematics.