P-adic numbers
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Notice: Incomplete
Prerequisites[edit | edit source]
Prime numbers
The Problem with Decadic Numbers[edit | edit source]
Decadic numbers have an issue in which there's a number which can be infinitely divided by 5. The problem with that is that you can't divide numbers divisible by 2 or 5 (the factors of 10) normally, and you have to eliminate the factors first. Such a number can't have all of its factors of 5 eliminated, so let's just remove the problem!
The Solution[edit | edit source]
Let's replace the number base with a prime number. This way, its only prime factor is the base itself. In order for a number to be infinitely divisible by its base, it would need to have an infinite number of trailing zeros, which would just make it 0, which already can't be divided by in the first place.
Interesting Properties[edit | edit source]
5-adic numbers have a square root of -1.
| 5-adic representation | (number)^2 (5-adic representation) |
|---|---|
| 2 | 4 |
| 112 | 144 |
| 1014212 | 100444 |
| 2213013121212 | 2024444 |
| 11232431203131212 | 2040344444 |
| 302104021134042203431212 | 413221444444 |
| 322331343011143322442212431212 | 13244124444444 |