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

[fresh_42]

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...

Continue reading...

 

[Office_Shredder]

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?

 

[fresh_42]

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.

 

[WWGD]

IIRC, p-adics are also a metric completion of the (Edit): Rationals?

 

[fresh_42]

IIRC, p-adics are also a metric completion of the Reals?

Yes, that's right.

 

[WWGD]

Yes, that's right.

Thanks. Does that play a role in your presentation here? Please don't mind my unhealthy preocupation with Mathematical minutiae.

 

[fresh_42]

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.

 

[martinbn]

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.