p-adic integers

[Bob Brown MSEE]

Can someone help me understand why the p-adic field is restricted to prime numbers. More specifically, prime radix for p-adic integers.

In link = #17 , I attempted to re-define the p-adic numbers in such a way that it ONLY included the rational numbers (of course base 10 however). The scientific notation was used to simplify notation. If valid, It results in a fun way to write repeating decimals without need for a minus sign.

Example: \(\displaystyle \frac{230}{7}\text{ = }\overline{571428}\text{.9}\) E 2

My guess was that no consistent definitons for +, *, mult-inverse, and add-inverse would be possible. But if I just use grade school digit manipulations, these seem to satisfy the field axioms for both addition and multiplication and is a commutative division algebra.

The validation of the field postulates using grade school arithmetic is even easier for these "repeating integer expressions" than for conventional "repeating decimal expressions". (because calculations are right-to-left, LSD first means that "carry" issues are solved)

 

[Bob Brown MSEE]

 

[Bob Brown MSEE]

Don't these Reversed integers appear to be infinite?

I have received an email with several excellent questions.
I'll try to cover them all here...

The repeating integer scientific notation (In link = #17) does appear to be infinite.
That is what makes them fun.

I have proposed a way to map this familiar looking notation to the rationals. That is not interesting, that can be done with integers. What is interesting is that these numbers look so like repeating decimals, and use the same arithmetic.

INFINITY: The interpretation of these numbers IN NO WAY conflicts with the notation called repeating decimals. Everyone is likely to agree that if they are interpreted as infinite -- then they have no conflict with repeating decimals and are undefined. I am defining them. They do in no way represent a sequence, nor the limit of a sequence. Further, I believe that it would be sound to allow a non-canonical form that has repeating digits on BOTH ends. I have used thousands of special cases and found no conflicts computationally.

Field: Since proposing these and doing some research, I am confident that they do form a legitimate subset of the p-adic field if I use binary numbers. I am not sure if there is a problem for base 10 (not prime). That is my question.

 

[JeffM]

I have received an email with several excellent questions.
I'll try to cover them all here...

The repeating integer scientific notation (In link = #17) does appear to be infinite.
That is what makes them fun.

I have proposed a way to map this familiar looking notation to the rationals. That is not interesting, that can be done with integers. What is interesting is that these numbers look so like repeating decimals, and use the same arithmetic.

INFINITY: The interpretation of these numbers IN NO WAY conflicts with the notation called repeating decimals. Everyone is likely to agree that if they are interpreted as infinite -- then they have no conflict with repeating decimals and are undefined. I am defining them. They do in no way represent a sequence, nor the limit of a sequence. Further, I believe that it would be sound to allow a non-canonical form that has repeating digits on BOTH ends. I have used thousands of special cases and found no conflicts computationally.

Field: Since proposing these and doing some research, I am confident that they do form a legitimate subset of the p-adic field if I use binary numbers. I am not sure if there is a problem for base 10 (not prime). That is my question.

This has to be off-topic because I know nothing about p-adic integers, but it is related to the topic that seems to have led to this thread. Norman Wildberger, who seems to be attempting to build up all of mathematics from the rational numbers while excluding all infinite processes and limits, defines "decimal numbers" as a subset of the rational numbers, namely those that have a denominator of any non-negative power of 2 or 5 or product of such powers. A decimal numeral (he calls it decimal form) is the normal decimal expansion of such a number. Every such expansion obviously has a finite number of terms. By definition then, rational numbers that are not "decimal numbers" are not represented by a "decimal numeral," and infinitely repeating decimals simply do not arise.

 

[Bob Brown MSEE]

Norman Wildberger, who seems to be attempting to build up all of mathematics from the rational numbers while excluding all infinite processes and limits, defines "decimal numbers" as a subset of the rational numbers, namely those that have a denominator of any non-negative power of 2 or 5 or product of such powers. A decimal numeral (he calls it decimal form) is the normal decimal expansion of such a number. Every such expansion obviously has a finite number of terms. By definition then, rational numbers that are not "decimal numbers" are not represented by a "decimal numeral," and infinitely repeating decimals simply do not arise.

Hmm. Sound's very interesting. Not sure why he would find such a definition useful.
I found his website, but he has almost 400 vids in a dozen different "Courses"
Do you have a link -- or remember which "Course" name? Thanks very much!

 

[JeffM]

Hmm. Sound's very interesting. Not sure why he would find such a definition useful.
I found his website, but he has almost 400 vids in a dozen different "Courses"
Do you have a link -- or remember which "Course" name? Thanks very much!

It starts at vid MF 66 under foundations of mathematics. As you say, he has hundreds of vids, and I have watched only a few. Thank you for reminding me about him. Here is what I think he is trying to do: replicate all of modern math on the basis of algebra and rational numbers, without reference to limits, infinities, functions, etc. So he has to talk about decimals because they are part of the mathematical landscape, and the way he does so simply limits decimal forms to finite forms, which is after all we ever encounter outside our imaginations. Decimals are quite peripheral to his attempt to develop calculus etc on rational numbers. I do not see that the effort required to force math into consistency with empirical reality is necessary (because as I have explained to you several times, I believe the reals, transfinites, etc are idealizations that can be thought about logically even though they correspond to nothing in the empirical universe.) But his approach does base math on entities that are empirically observable, and that seems to have a number of positive aspects. Math his way ends up looking very odd to me, but that is because I have spent almost 70 years getting used to math in its usual presentation.

 

[Bob Brown MSEE]

Math his way ends up looking very odd to me, but that is because I have spent almost 70 years getting used to math in its usual presentation.

Thank you so much, I'll certainly check it out. I have not had as many years of experience as you. But I am approaching infinity too, hopefully asymptotically (for both of us)