Mathematicians Use Numbers Differently From The Rest of Us

TL;DR

p-adic numbers, a different number system that includes infinite digits going off to the left of the decimal point, are a powerful tool in mathematics and have applications in solving complex equations and mathematical problems.

Transcript

• Take the number 5 and square it, you get 25. Now take 25 and square it, you get 625. Square 625, and you get 390,625. Do you see the pattern? 5 squared ends in a 5, 25 squared ends in 25, and 625 squared ends in 625. So does this pattern continue? Well, let's try squaring 390,625. It doesn't quite end in itself, but the last 5 digits match, so it... Read More

Key Insights

• #️⃣ p-adic numbers have unique properties that make them a powerful tool in mathematics, particularly in solving equations and mathematical problems that are difficult to solve using ordinary numbers.
• 🥘 The geometric series formula can be used to find solutions to equations involving p-adic numbers, allowing for the calculation of the sum of an infinite geometric series despite its apparent divergence.
• #️⃣ The notion of size in p-adics is different from that in ordinary numbers, with larger numbers actually being considered smaller. This unique perspective opens up new possibilities in solving equations and understanding mathematical concepts.

Questions & Answers

Q: What are p-adic numbers and how are they different from ordinary numbers?

p-adic numbers are a number system that includes infinite digits going off to the left of the decimal point. Unlike ordinary numbers, p-adics have different notions of size, with larger numbers actually being considered smaller.

Q: How are p-adic numbers useful in solving equations?

p-adic numbers have unique properties that allow them to be added, subtracted, and multiplied like ordinary numbers. They can also be used to find rational solutions to equations and solve complex mathematical problems that are difficult to solve using ordinary numbers.

Q: How do p-adic numbers relate to Diophantus' "Arithmetica" and Fermat's Last Theorem?

p-adic numbers were used by mathematicians to solve equations similar to those discussed by Diophantus in "Arithmetica." p-adic numbers were also utilized in the proof of Fermat's Last Theorem, providing a new approach to solving complex equations.

Q: How does the geometric series formula work in p-adic numbers?

The geometric series formula can be used to find solutions to equations involving p-adic numbers. This formula allows for the sum of an infinite geometric series to be calculated, even if the sum seems to diverge based on conventional notions of size.

Summary & Key Takeaways

• p-adic numbers are a number system that includes infinite digits going off to the left of the decimal point, and they have unique properties that make them useful in solving complex equations.

• These numbers can be added, subtracted, and multiplied, just like ordinary numbers, and they also contain rational numbers and negative numbers.

• The geometric series formula can be used to find solutions to equations involving p-adic numbers, and the notion of size in p-adics is different, with larger numbers actually being considered smaller.