The 5-adic metric for the rational numbers.

[robm447]

I was the best in class at maths, but I gave up at 17. My close friend went to uni to study it and he has recently met with all manner of problems and is so frustrated, he's thinking of leaving. I was able to help him with some trig and calculus stuff as I had the basics and some advanced knowledge so that he understands it now, but I never really started on number theory. He sent me some questions he's been asked to do and I have spent many hours watching youtube videos about p-adic numbers and the more I watched, the more inadequate I felt. I wonder if some very kind person could help me and my friend out by offering suggestions/solutions for the following. The questions were sent to me as an image and I don't have the wherewithal to convert it into text for the forum, so I've made the image available via shoebox. Thanks in advance for anyone could offer assistance. Rob

https://secure.shoeboxapp.com/photo...a6ef47e8666b1a40bdf0a396c2f99479fcf25b499f402

 

[stapel]

My close friend...sent me some questions he's been asked to do...and I don't have the wherewithal...

https://secure.shoeboxapp.com/photo...a6ef47e8666b1a40bdf0a396c2f99479fcf25b499f402

Most of the helpers here have learned, from hard experience, that attempting to tutor through a "translator" who "doesn't speak the language" is not likely to end well.

Kindly please have your friend reply here with a clear listing of his thoughts and efforts so far. When he replies, please have him include clarification of what "(M1)", "(M2)", and "(M3)" are, and what are the "results from Book A" that he thinks are likely to be useful here. Thank you!

For other viewers:

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February 24, 2016
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Question 2 - 18 marks


In this question, we introduce a metric for the rational numbers, known as the 5-adic metric, that is very different to the usual one for the rationals that is obtained from the Euclidean metric for the reals. You should be able to do this question after reading Chapters 14 and 15.

Using results from Book A, one can show that for each non-zero rational number, x say, there is a unique integer m such that:

. . . . .\(\displaystyle x\, =\, 5^m\, \dfrac{a}{b}\)

where a is a non-zero integer, b is a natural number, and 5 does not divide either of a and b. We use this fact to define a function f₅ from the rationals to the reals by:

. . . . .\(\displaystyle f_5(x)\, =\, \begin{cases}5^{-m} &,\, \mbox{ if }\, x\, =\, 5^m\, \dfrac{a}{b}\, \mbox{ and } 5\, \nmid \, a,\, b \\ 0 &,\, \mbox{ if }\, x\, =\, 0 \end{cases}\)

(a) Write down f₅(1), f₅(5), and f₅(5ⁿ) for natural numbers n.

(b) Show that f₅ has the following properties:

. . .i. For each rational x, f₅(x) > 0, with equality if, and only if, x = 0.

. . .ii. For each rational x, f₅(-x) = f₅(x).

. . .iii. For rationals x and y, f₅(x + y) < max{ f₅(x), f₅(y) }.

Now define a distance function d₅ from the set of rational-valued points to the reals by:

. . . . .\(\displaystyle d_5(r,\, s)\, =\, f_5(r\, -\, s)\)

(c) Using the results of part (b), or otherwise, verify that (M1), (M2), and (M3) hold, and hence show that d₅ is a metric for the rationals.

(d) Show that:

. . . . .\(\displaystyle \left(\, 5^n\, \right)_{n\, 1}^{\infty}\)

d₅-converges to 0.