The Distance Between Numbers - Numberphile

TL;DR

A sequence with increasing 9s converges unexpectedly to -1 using the 2-adic metric.

Transcript

If you were to define what it meant to be  a banana, you had to tick every box on that   list and then you could be nothing but a banana.  We're going to look at a sequence and show that   it converges to a limit you weren't expecting. So  the sequence is as follows: the first term is 9, second term is 99, third term 999.  Fourth term - you guessed... Read More

Key Insights

• 📈 The sequence of increasing 9s converges to -1 unexpectedly using the 2-adic metric.
• ❓ Understanding the mathematical definition of distance is crucial in analyzing the convergence of sequences.
• 💨 The 2-adic metric provides a unique way to measure distance and convergence in number sequences.
• 👮 Axioms of distances, including positivity, symmetry, and the triangle law, are essential in defining a distance function.
• ⛔ Different sequences can have varying convergence properties, and the specific numbers in the sequence influence the limit it converges to.
• 📈 The 2-adic metric can be generalized to p-adic metrics where p is a prime number, offering flexibility in distance measurement.
• 😆 Mathematical rigor is needed to verify that a metric satisfies the necessary conditions to be considered a valid distance function.

Questions & Answers

Q: How does the sequence of numbers composed of increasing 9s converge to -1?

The sequence converges to -1 by demonstrating that the distance between the nth term and -1 tends to zero as n approaches infinity using the 2-adic metric.

Q: Why is the 2-adic metric used to measure distance in this sequence?

The 2-adic metric is used because it allows for the distance between the numbers in the sequence to approach zero as the numbers get larger, unlike the real number metric where the distance would be infinity.

Q: Can the convergence to -1 be replicated with sequences using different numbers?

Yes, the convergence to a specific limit depends on the sequence of numbers used, and changing the numbers can result in different convergence or even non-convergence situations.

Q: How does the sequence with increasing 8s differ in convergence from the sequence of increasing 9s?

Changing the numbers in the sequence alters the convergence properties, and the sequence with increasing 8s may converge to a different limit or not converge at all compared to the sequence with increasing 9s.

Summary & Key Takeaways

• The sequence of numbers 9, 99, 999, etc., appears to diverge to infinity, but it actually converges to -1.

• Convergence is demonstrated using the 2-adic metric, where the distance between terms tends to zero.

• The mathematical definition of distance, axioms of distances, and the triangle law are explained in detail.