I understand everything I've read about p-adic

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I understand everything I've read about p-adic numbers but I can't internalize any motivation

Diophantine equations, lifting, strong triangle inequality, two numbers are closer if their difference is highly divisible by p, fractal towers, completion (filling holes in the rationals by representing decimals in a p-adic base).

Please. Help.
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I understand everything I've read about p-adic numbers but…

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