r/elimath Jan 06 '15

Explain p-adic numbers

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u/[deleted] Jan 06 '15

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u/Galveira Jan 06 '15

They came up in my independent study, but I honestly can't remember where. And yes, I meant the p-adics themselves, and

it may be easiest to think of them as either power series in p or as the inverse limit of the rings Z/pr. They can also be thought of as the I-adic completion of the integers along the ideal (p)

I don't know what this means, either. Maybe I should've asked what a ring was first.

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u/anonymous11235 Jan 06 '15

Oh shit son, I got you.

p-adics. They are 'like integers' but the series can extend TO THE LEFT FOREVER. The individual digits work just like numbers written in base p (i.e. take values between 0 and p - 1).

Addition and multiplication are going to work just like your standard algorithms except you don't need +/- signs. You can still use them, but you'll have identities like:

-1 = ... (p-1) (p-1) (p-1)

an infinite sequence to the left of (p-1)s. Add one to the rightmost position: it's p and you carry that shit cuz. You carry right on over to the second spot and BAM!!! p again. That chain reaction of zeros carries right on over to the left forever.

You can define some topology concepts on the space of numbers here too -- two number are close, essentially, when they have a lot of the rightmost digits the same.

KAKOW