Counting to p-adic Calculus: All Number Systems That We Have

In summary, the history of numbers is a long and complex one, spanning from ancient Babylon and India to modern mathematicians such as Kurt Hensel. The concept of zero, while often romanticized, was originally used as a placeholder for empty spaces in accounting. The digits 0 to 9 were first introduced in Sanskrit and have played a crucial role in the development of our number systems. Additionally, p-adic numbers, which are a metric completion of the rationals, have been important in areas such as number theory and homological algebra.
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fresh_42
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An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha’far Muhammad ibn Musa al-Chwarizmi (##\sim ## 780 – 845), Gerbert of Aurillac aka pope Silvester II. (##\sim ## 950 – 1003), Leonardo da Pisa Fibonacci (##\sim## 1170 – 1240), Johann Carl Friedrich Gauß (1777 – 1855), Sir William Rowan Hamilton (1805 – 1865), to Kurt Hensel (1861 – 1941). This would lead too far. Instead, I want to consider the numbers by their mathematical meaning. Nevertheless, I will try to describe the mathematics behind our number systems as simple as possible.
I like to consider the finding of zero as the beginning of mathematics: Someone decided to count what wasn’t there! Just brilliant! However, the truth is as often less glamorous. Babylonian accountants needed a placeholder for an empty space for the number system they used in their books. The digits zero to nine have been first introduced in India. In Sanskrit, zero stands for emptiness...

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This is great, but your perturbation is digits to prove non countability of ##\mathbb{R}## walks right into the fact that a number has more than one representation. I think if you iterate by 3 instead of 1 you avoid this, but maybe that's too finicky and not worth it for an overview.

Edit: thinking a little more, as long as you insist on writing in your initial list the representation that has the largest digit as early as possible you avoid the issue?
 
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Office_Shredder said:
This is great, but your perturbation is digits to prove non countability of ##\mathbb{R}## walks right into the fact that a number has more than one representation. I think if you iterate by 3 instead of 1 you avoid this, but maybe that's too finicky and not worth it for an overview.

Edit: thinking a little more, as long as you insist on writing in your initial list the representation that has the largest digit as early as possible you avoid the issue?
Corrected from the idea in my mind to the proof in my book to the expense of more technical babble which I tried to avoid.
 
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IIRC, p-adics are also a metric completion of the (Edit): Rationals?
 
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WWGD said:
IIRC, p-adics are also a metric completion of the Reals?
Yes, that's right.
 
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fresh_42 said:
Yes, that's right.
Thanks. Does that play a role in your presentation here? Please don't mind my unhealthy preocupation with Mathematical minutiae.
 
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WWGD said:
Thanks. Does that play a role in your presentation here? Please don't mind my unhealthy preocupation with Mathematical minutiae.
They are a metric completion. I think we cannot say "of the reals" since they are not included in ##\mathbb{Q}_p.## ##p##-adic analysis is strange. The topological idea behind completion is easy: we have an evaluation, that defines a metric, so we have Cauchy-sequences and ##\mathbb{Q}_p## just gathers all limits per definition.

They are interesting for homological algebra since they can be introduced via projective limits.
 
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fresh_42 said:
They are interesting for homological algebra since they can be introduced via projective limits.
Not sure about this, but they are certainly important in number theory.
 

1. What is p-adic calculus?

P-adic calculus is a mathematical system that extends the concept of real numbers to include numbers that are infinitely large or infinitely small. It is based on the idea of p-adic numbers, which are numbers that can be written as a power of a prime number, p.

2. How is p-adic calculus different from traditional calculus?

P-adic calculus differs from traditional calculus in several ways. First, it deals with numbers that are not limited to a specific range, unlike real numbers. Additionally, p-adic calculus has its own set of rules and operations, which are different from those of traditional calculus. Lastly, p-adic calculus is more suitable for analyzing discrete and non-continuous systems, while traditional calculus is better suited for continuous systems.

3. What are the applications of p-adic calculus?

P-adic calculus has various applications in mathematics, physics, and computer science. It is used in number theory, cryptography, and coding theory. In physics, p-adic calculus is used to study quantum mechanics and string theory. It also has applications in computer science, particularly in the design of error-correcting codes.

4. Is p-adic calculus difficult to learn?

P-adic calculus can be challenging to learn for those who are not familiar with abstract algebra and number theory. It requires a strong foundation in mathematics, particularly in algebra and analysis. However, with proper study and practice, it can be understood by anyone with a strong mathematical background.

5. How is p-adic calculus related to other number systems?

P-adic calculus is a generalization of the real numbers and includes other number systems such as rational numbers, integers, and complex numbers. It also has connections to other mathematical structures, such as fields, rings, and groups. P-adic calculus provides a unique perspective on these number systems and allows for new insights and applications.

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