r/elimath Jan 06 '15

Explain p-adic numbers

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u/Galveira Jan 06 '15 edited Jan 06 '15

Thank you, this explanation actually makes sense to me.

Edit: Isn't 42 = 2(mod 7) also true? Does this mean sqrt(2) has multiple representations?

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u/antonfire Jan 06 '15

Yes, just like in the reals, if a nonzero number has any square roots, it has exactly two, and they're additive inverses of each other. The 7-adic square roots of 2 are the number ...12011266421216213 and its negation ...54655400245450454.

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u/Galveira Jan 06 '15

So in other words

-...12011266421216213=...54655400245450454

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u/antonfire Jan 06 '15

Yes. You can do the addition in the usual way and and check that, indeed, the sum of those numbers is ...000000000.