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Math Help - P-adic numbers

  1. #1
    Super Member Deadstar's Avatar
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    P-adic numbers

    Having looked through my entire lecture notes section entitled p-adic numbers, I have concluded that there is not one mention of the word adic in the whole section. Hence, I have no idea how I am supposed to do this.

    Calculate 1/3 as a 5-adic number

    After wikipediaing, I have realized that it should look something like...

    a_0 + a_1 5 + a_2 + 5^2 + a_3 5^3 + \dots

    But I have no idea how to get there.

    Things we did cover in the section were,

    Principle of Domination
    Solving x^2 \equiv a(\mod p^n)
    Non Archemedean Values

    Are any of these going to help with this question? I can't see how.
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  2. #2
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    Quote Originally Posted by Deadstar View Post
    Having looked through my entire lecture notes section entitled p-adic numbers, I have concluded that there is not one mention of the word adic in the whole section. Hence, I have no idea how I am supposed to do this.

    Calculate 1/3 as a 5-adic number

    After wikipediaing, I have realized that it should look something like...

    a_0 + a_1 5 + a_2 + 5^2 + a_3 5^3 + \dots

    But I have no idea how to get there.

    Things we did cover in the section were,

    Principle of Domination
    Solving x^2 \equiv a(\mod p^n)
    Non Archemedean Values

    Are any of these going to help with this question? I can't see how.

    \left|\frac{1}{3}\right|_{5}=1\Longrightarrow \frac{1}{3}\in\mathbb{Z}_5= the 5-adic integers \Longrightarrow \left[\frac{1}{3}\right]_5=a_0+5a_1+5^2a_2+\ldots\,,\,\,0\leq a_i<5 \Longrightarrow \frac{1}{3}=a_0\!\!\!\pmod 5\Longrightarrow a_0=2\,,\,\,\frac{1}{3}-2=5a_1\!\!\!\pmod {5^2} \Longrightarrow -\frac{5}{3}=5a_1\!\!\!\pmod{5^2}\Longrightarrow -\frac{1}{3}=a_1\!\!\!\pmod 5 \Longrightarrow a_1=3\Longrightarrow \frac{1}{3}-2-3\cdot 5=5^2a_2\!\!\!\pmod{5^3}\Longrightarrow -\frac{50}{3}=5^2a_2\!\!\!\pmod {5^3} \Longrightarrow -\frac{2}{3}=a_2\!\!\!\pmod 5\Longrightarrow a_2=1 , and etc...

    So \left[\frac{1}{3}\right]_5=2+3\cdot 5+1\cdot 5^2+\ldots

    Tonio
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  3. #3
    Super Member Deadstar's Avatar
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  4. #4
    Super Member Deadstar's Avatar
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    Here was the given solution: Now 3|(5^2 - 1) = 24, so

    1/3 = 8/24 = -8/(1 - 5^2) = -8(1 + 5^2 + 5^4 + ...

    = -(3 + 5^1 + 3\cdot5^2 + 5^3 + 3\cdot5^4 + 5^5 +... )

    = 2 + 3\cdot5^1 + 5^2 + 3\cdot5^3 + 5^4 + 3\cdot5^5 + ...

    Times like these I wonder why my uni is ranked so highly...
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  5. #5
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    Quote Originally Posted by Deadstar View Post
    Here was the given solution: Now 3|(5^2 - 1) = 24, so

    1/3 = 8/24 = -8/(1 - 5^2) = -8(1 + 5^2 + 5^4 + ...

    = -(3 + 5^1 + 3\cdot5^2 + 5^3 + 3\cdot5^4 + 5^5 +... )

    = 2 + 3\cdot5^1 + 5^2 + 3\cdot5^3 + 5^4 + 3\cdot5^5 + ...

    Times like these I wonder why my uni is ranked so highly...
    If it is Ediburgh's then it justly has a good name, though I can't tell how its maths dept. is and even less its algebra strength.

    Anyway, the solution given is a very good one, but not always is that easy to find such a nice relation between a given number (1/3, in this case) and some series in powers of 5 ...

    Tonio
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  6. #6
    Super Member Deadstar's Avatar
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    Quote Originally Posted by tonio View Post
    If it is Ediburgh's then it justly has a good name, though I can't tell how its maths dept. is and even less its algebra strength.

    Anyway, the solution given is a very good one, but not always is that easy to find such a nice relation between a given number (1/3, in this case) and some series in powers of 5 ...

    Tonio
    so, let me see if this is right...

    In p-adic numbers, 1 + p + p^2 + p^3 + \dots = \frac{1}{1-p}?

    And a_0 = 2 since that is the multiplicative inverse of \frac{1}{3} in mod 5?

    As for Edinburgh... It's not that we have bad facilities or bad research or whatever... But look at any university rankings list and you'll see Edinburgh will have one of the lowest student satisfaction ratings in probably the top 100.

    I can't speak for the rest of the courses (although I know many friends were not happy with the teaching, friends who graduated with 2:1s and firsts) but I know personally the quality of the lectures and the general organization/management of courses in the maths dept are way below par.

    With about 90% of my course here's how it works...

    Lecture, write 8-12 sides of A4 paper.
    Half an hour later, full notes for the course appear on the website...
    ...
    What's the point in the lectures?
    You're constantly writing so it's very difficult to understand things as you are immediately writing down something new (even the uni says, just get it down, understand it later).

    Just an example but in general it's not Edinburghs fault. I firmly believe that maths lectures are borderline pointless. Someone who excels in their field (i.e the lecturer) should not be used to simply regurgitate notes to students.
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  7. #7
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    Quote Originally Posted by Deadstar View Post
    so, let me see if this is right...

    In p-adic numbers, 1 + p + p^2 + p^3 + \dots = \frac{1}{1-p}?

    And a_0 = 2 since that is the multiplicative inverse of \frac{1}{3} in mod 5?


    Indeed so, both.


    As for Edinburgh... It's not that we have bad facilities or bad research or whatever... But look at any university rankings list and you'll see Edinburgh will have one of the lowest student satisfaction ratings in probably the top 100.

    I can't speak for the rest of the courses (although I know many friends were not happy with the teaching, friends who graduated with 2:1s and firsts) but I know personally the quality of the lectures and the general organization/management of courses in the maths dept are way below par.

    With about 90% of my course here's how it works...

    Lecture, write 8-12 sides of A4 paper.
    Half an hour later, full notes for the course appear on the website...
    ...
    What's the point in the lectures?
    You're constantly writing so it's very difficult to understand things as you are immediately writing down something new (even the uni says, just get it down, understand it later).

    Just an example but in general it's not Edinburghs fault. I firmly believe that maths lectures are borderline pointless. Someone who excels in their field (i.e the lecturer) should not be used to simply regurgitate notes to students.

    It really sounds nasty, though listening and writing are supposed to aid in the understanding, but 8-12 sides of A4 paper?? What is that, humanities!?

    Anyway, checking on the net Edinburgh's Univ. has a good name in fields not directly related to maths (chemistry, physics, literature,...), so perhaps the maths dept. is not their best one.

    Tonio
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  8. #8
    Super Member Deadstar's Avatar
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    Quote Originally Posted by tonio View Post
    It really sounds nasty, though listening and writing are supposed to aid in the understanding, but 8-12 sides of A4 paper?? What is that, humanities!?

    Anyway, checking on the net Edinburgh's Univ. has a good name in fields not directly related to maths (chemistry, physics, literature,...), so perhaps the maths dept. is not their best one.

    Tonio
    (Long post sorry...)

    Maybe not, I'm not sure about the rest.

    I just believe that there is a better way to be learning maths. If I get stuck on something (and I mean, a section of the course not just a question or single theorem), then I go see a lecturer about it and within half an hour of him explaining it I will understand an entire tutorial.

    For example, I couldn't get my head around Ito's formula and stochastic differential equations so I went to see the lecturer last week (this made up about 15% of the course material wise I would say). He spent 10 minutes explaining it to me and the way he explained it was to deconstruct it right down and talk to me like I hadn't seen it before.

    This method of teaching, I find, works really well and I came out of it able to do almost the entire tutorial straight away. Of course I still found the harder questions difficult but my understanding of the material was sooooo much greater than when I came out the lecturers not having digested any of it.

    Now imagine if lectures were done in such a way that everyone came out of it and were immediately able to settle down and barge through a large part of the tutorial instead of having to re-read the notes many times and read books and check wikipedia etc... This seems like a much faster way of learning and gives folk more time to master the material rather than learn it.

    A simple idea would be to give out lecture notes the week before lectures, one weeks worth at a time. Then there would be two lectures during the week which explain things clearly and in detail. There's no need to write down every theorem, definition and proof, you will already have them in front of you. But what you could write down is the fine details of each proof and theorem. "Why can we do this"? "Here's a few examples". "Let's explain this a bit better", etc...

    Simple example would be...
    (see attached pic). That would have been given out in the notes the week before the lectures.
    In the lectures, a minute could be used just to perhaps, draw one of those injection/surjection diagrams (the oval things with dots in them) to show exactly how this g(x) came about.

    This is just a example using a simple theorem but harder theorems would benefit from more in depth study in the lectures...


    Done...
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