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sniffnoyI think of p-adic numbers backwards.
That is, I think of them as extending from left to right -- first the ones place, then the p's place, etc. I don't get mixed up about how they work (except occasionally forgetting that they are characteristic 0, not characteristic p, or equivalently forgetting to carry); I know which places are which. When adding them, I carry as going to the right, I don't forget and carry to the left. I think of the 1s place as just to the right of the decimal point, not just to the left of it.
Of course, thinking p-adically, this makes sense, as it means I'm putting the p-adically most significant digit first. And it's damned inconvenient to write numbers that start with an ellipsis. But it's backwards from how whole numbers are normally written (I think the one thing that occasionally does trip me up with this is when I have to consider whole numbers embedded in the p-adics, and actually think of them as whole numbers, not just finite strings of digits), and it's backwards from how everyone else writes them.
The reason I mention this is because it caused a slight problem on my homework last night when I referred to the "first digit" of a p-adic number, and later I realized that everyone else would think of it as the last digit, so I had to change that. Perhaps I should just say most significant or least significant? But even speaking of the "next digit" is problematic; I don't think there's a direction unambiguous substitute for "next" and "previous". And really, it's possible that most significant and least significant could also be misinterpreted, since normally they mean the other thing!
Hm.
In any case, here's the homework problem (due Monday, so even though I've done it, I'd prefer not discussing solutions here till then). The class is analytic number theory; Lagarias put these two problems on the problem set but marked them as optional and off-topic. (So naturally I did them close to first.) The second was also marked as difficult, but seeing as I'm used to things only being marked difficult if they're nigh-impossible, it was surprisingly easy. I mean, it wasn't easy at all, but it wasn't nigh-impossible, is my point, because I completed it.
The first is just a bit of calculus: Say you're given R>0, and you pick two real numbers a, b with |a|,|b|≤R uniformly at random. Let P(R) be the probability that the polynomial x²+ax+b can be factored over the real numbers. What is the limit as R→∞ of P(R)? What is the limit as R→0 of P(R)? And what is P(1)?
The second is exactly the same, except throughout replace "real numbers" with "p-adic numbers". (Note that some of the limits may not exist.)
Surprisingly, the p=2 case is only a little harder than the p≠2 case! (Also, I'm pretty sure I could compute P in general, but I just didn't bother because it wasn't called for and it just seems like a nuisance.)
-Harry
That is, I think of them as extending from left to right -- first the ones place, then the p's place, etc. I don't get mixed up about how they work (except occasionally forgetting that they are characteristic 0, not characteristic p, or equivalently forgetting to carry); I know which places are which. When adding them, I carry as going to the right, I don't forget and carry to the left. I think of the 1s place as just to the right of the decimal point, not just to the left of it.
Of course, thinking p-adically, this makes sense, as it means I'm putting the p-adically most significant digit first. And it's damned inconvenient to write numbers that start with an ellipsis. But it's backwards from how whole numbers are normally written (I think the one thing that occasionally does trip me up with this is when I have to consider whole numbers embedded in the p-adics, and actually think of them as whole numbers, not just finite strings of digits), and it's backwards from how everyone else writes them.
The reason I mention this is because it caused a slight problem on my homework last night when I referred to the "first digit" of a p-adic number, and later I realized that everyone else would think of it as the last digit, so I had to change that. Perhaps I should just say most significant or least significant? But even speaking of the "next digit" is problematic; I don't think there's a direction unambiguous substitute for "next" and "previous". And really, it's possible that most significant and least significant could also be misinterpreted, since normally they mean the other thing!
Hm.
In any case, here's the homework problem (due Monday, so even though I've done it, I'd prefer not discussing solutions here till then). The class is analytic number theory; Lagarias put these two problems on the problem set but marked them as optional and off-topic. (So naturally I did them close to first.) The second was also marked as difficult, but seeing as I'm used to things only being marked difficult if they're nigh-impossible, it was surprisingly easy. I mean, it wasn't easy at all, but it wasn't nigh-impossible, is my point, because I completed it.
The first is just a bit of calculus: Say you're given R>0, and you pick two real numbers a, b with |a|,|b|≤R uniformly at random. Let P(R) be the probability that the polynomial x²+ax+b can be factored over the real numbers. What is the limit as R→∞ of P(R)? What is the limit as R→0 of P(R)? And what is P(1)?
The second is exactly the same, except throughout replace "real numbers" with "p-adic numbers". (Note that some of the limits may not exist.)
Surprisingly, the p=2 case is only a little harder than the p≠2 case! (Also, I'm pretty sure I could compute P in general, but I just didn't bother because it wasn't called for and it just seems like a nuisance.)
-Harry
