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p-adic numbers: A minithread
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There are many people here more qualified than me to write this, and most of my audience here understand this better than me anyway, but here's my attempt at a simple explanation of the p-adic numbers.
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Okay, so let's consider the real numbers first. And we do that by considering the rational numbers first. Arrange the rational numbers from smallest (most negative) to biggest (most positive). This is the good old number line. Except there are "gaps".
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What do we mean by gaps? Well we can consider an infinite sequence of rational numbers whose entries become closer and closer to each other, and yet, this sequence does not converge to a rational number. This is our "gap", and by filling in the gaps we have the "real line"
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The real numbers are mindbreaking to me to be honest. The rationals are kind of less so. That's why I just think of the "rational number line" with gaps and the other real numbers simply fill those gaps. This mental trick makes it easier for me, kind of.
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Okay, now the p-adic numbers, for p some prime number. Once again, our mental trick is to make the rational numbers the star of the show. So consider only the rational numbers first. But now we're going to arrange them in a strange way.
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Just like in life, there are many notions of closeness. There is geographic closeness, for example. But in another sense I could be "closer" to my family and friends back home, more than 8,000 miles away, than to some stranger here. We shall apply this to numbers.
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We now say two rational numbers are close to each other if their difference is divisible by a big power of p. So if p=5 for example, now 1 is closer to 6 than to 2. 1 is even closer to 126 than either 2 or 6. 1 is as close to 2 as it is close to 3 however.
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I don't think they look like a line in this arrangement anymore, but we still have a notion of "closeness". This means we can talk about converging sequences where the entries become "closer" in our new sense (their difference is divisible by bigger powers of p), and gaps!
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Just like how we constructed the reals by filling in gaps (sequences that don't converge to a rational), with our ordinary notion of closeness, with our new "p-adic" notion of closeness there will be gaps, and filling them in is what gives us the p-adic numbers!
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