Liouville's number, the easiest transcendental and its clones (corrected reupload)

TL;DR

Liouville's number is a transcendental number with a unique decimal expansion pattern, and it can be used to create a clone of the real numbers within the real numbers.

Transcript

Welcome to another Mathologer video. Liouville's number the monster up there consists of infinitely many isolated islands of 1s at the 1! th, 2! th, 3! th, etc. digits with exploding gaps of zeros between them. As I promised you at the end of the last video, today's mission is to show you a nice visual way of seeing that this number is transcendent... Read More

Key Insights

• 👍 Liouville's number is transcendental and its transcendence can be proven through an examination of the digit patterns of its squared truncations.
• #️⃣ The decimal expansion of Liouville's number allows for the creation of a clone of the real numbers within the real numbers, consisting entirely of transcendental numbers.
• #️⃣ The clone of the real numbers based on Liouville's number has measure zero, indicating that it takes up no space within the real numbers.

Questions & Answers

Q: What is Liouville's number and how does its decimal expansion pattern make it transcendental?

Liouville's number is a transcendental number with a decimal expansion consisting of isolated islands of 1s and exploding gaps of zeros between them. This unique pattern of digits, with the increasing gaps between the 1s, contributes to its transcendence.

Q: How is the transcendence of Liouville's number proven?

The proof of Liouville's number being transcendental involves examining the digit patterns of the squared truncations of the number. It is demonstrated that, for each power of Liouville's number, there is a certain point from which all the digits of the truncations are correct, indicating its transcendence.

Q: Can Liouville's number be used to create a clone of the real numbers within the real numbers?

Yes, using Liouville's number as a template, it is possible to create a clone of the real numbers consisting entirely of transcendental numbers. This clone has the same cardinality as the set of real numbers but has measure zero, meaning it takes up no space within the real numbers.

Q: How is the decimal expansion of Liouville's number related to the creation of the clone of the real numbers?

The decimal expansion of Liouville's number, with isolated islands of 1s at specific digit locations, is used to create a clone of the real numbers by replacing the 1s with other digits. As long as infinitely many of these replacement digits are nonzero, the resulting numbers in the clone will also be transcendental.

Summary & Key Takeaways

• Liouville's number is a transcendental number with a decimal expansion consisting of isolated islands of 1s and exploding gaps of zeros between them.

• The proof of Liouville's number being transcendental is accessible to those with some exposure to real analysis, and it involves examining the digit patterns of the squared truncations of the number.

• Using Liouville's number as a template, it is also possible to create a clone of the real numbers consisting entirely of transcendental numbers, which has measure zero.