Has anyone heard of p-adic numbers? The topic came up briefly the other day in my Number Theory lecture. However, we were not in a position to discuss it in any detail.

I'd like to know more in the subject but the internet resources are fairly uninviting and scary. So I was wondering if anyone has encountered these numbers and associated structure. If so I would appreciate a introduction to them.

I have heard that you can use them to extend the rationals to the reals without the need for analysis. Much the same way as you can go from the natural numbers to the integers and then to the rationals using basic arithmetic. Normally, going from the rationals to the reals requires some fairly complicated analysis.

Also, why does p need to be prime?

Has anyone heard of p-adic numbers? The topic came up briefly the other day in my Number Theory lecture. However, we were not in a position to discuss it in any detail.

I'd like to know more in the subject but the internet resources are fairly uninviting and scary. So I was wondering if anyone has encountered these numbers and associated structure. If so I would appreciate a introduction to them.

there are the completion in the p-adic norm. do you understand the p-adic norm?


I have heard that you can use them to extend the rationals to the reals without the need for analysis.

no, that is not true in anyway. there is a way to define the p-adic numbers without mentioning analysis, but it has nothing to do with the reals, and is a complicated inverse limit statement that realy wouldn't be helpful to talk aobut if you don't see why p must be a prime below.

Much the same way as you can go from the natural numbers to the integers and then to the rationals using basic arithmetic. Normally, going from the rationals to the reals requires some fairly complicated analysis.

Also, why does p need to be prime?

otherwise it, the p-adic norm, isn't a well behaved valuation.