The p-adics

In a post on January 17th 2019 title The Golden Key, I wrote:

I came across the Golden Key when perusing Kumar Asok Mallik's book The Story of Numbers during his introduction to prime numbers on page 23. This is quite an interesting book that I've added to my Calibre library ... If I can read an entry a day from this book, I'll soon be a wiser man mathematically.

Well it's no surprise that I didn't read an entry a day and in fact I only stumbled upon the post and its reference to Mallik's book when searching for posts related to \(p\)-adic numbers (which he discusses in his book). There is a good article on the \(p\)-adics in Quanta Magazine titled An Infinite Universe of Number Systems. Figure 1 shows a visualisation of 3-adic numbers:

Figure 1

The diagram makes sense when the following diagrams are considered in Figures 2, 3 and 4. The tube-like structures arise when\(\mod{3^n}\) with \(n=1, 2, 3, ... \) is applied progressively to the natural numbers. The natural numbers become grouped in an entirely new way.

Figure 2

Figure 3

Figure 4

Looking at Figure 4, it's clear in this system that 37 is closer to 10 than it is to 36. As the quanta article explains it:

The size of a \(p\)-adic number is determined by the prevalence of \(p\) in its prime factorisation. Numbers with more \(p\)s are smaller. For example, in the 3-adics, \(486_{10}=200000_3\) is “small” because it has many 3s in its prime factorisation (486 = 2 x 3 x 3 x 3 x 3 x 3). Another way to think about size is to think about which numbers are close to 0. In the \(p\)-adics, integers are closer together when they share a room at higher levels of the tower. The numbers 0 and 486 share a room up to the fifth level, whereas 0 and 6 share a room on only the first level — indicating that 0 is closer to 486 than to 6 and thus 486 is smaller than 6.

Mallik uses the 7-adic number system for illustration purposes. Here he talks about negative numbers in such a system:

A surprising fact is that for \(p\)-adic integers we do not need any negative sign to indicate negative integers. We determine the negative of a positive integer by determining what needs to be added to this positive integer to yield zero, i.e., by subtracting this number from 0. For example 7-adic expansion of −1 can be written as an integer . . . 6 6 6 6 with infinitely many 6’s on the left. We can verify this result by adding 1 to this 7-adic integer: 

· · · 6 6 6 6 + · · · 0 0 0 1 = · · · 0 0 0 0

Thus we can write that \(-1 = 6  +6 \times 7 + 6 \times 7^2 + 6 \times 7^3 + ... \) which seems odd but if 1 is added to both sides we see that it is in fact true. He goes on to say that:

Now we show that some \(p\)-adic integers also represent rational fractions like, say, 1/2. The 7-adic integer · · · 3 3 3 4 with infinitely many 3’s represents 1/2, as can easily verified by multiplying this number by 2 or by adding this number to itself:

 · · · 3 3 3 4  x  2 = · · · 0 0 0 1  

Mallik goes on to mention that \( \sqrt{2} = · · · \text{ 6 2 1 3 } \) because:

· · · 6 2 1 3  x  · · · 6 2 1 3 = · · · 0 0 0 2

Figure 5 shows how 5-adic numbers can be equal to \( \sqrt{-1} \):

Figure 5: source

It's easy to get confused by the above and perhaps this paper is a better introduction to the mechanics of the topic as it details how to add, subtract, multiply and divide p-adic numbers. Furthermore, it points out the difference between the p-adic representation of a number and the p-ary. See Figure 6.

Figure 6: source

The p-adic numbers pop up everywhere in higher Mathematics. For example, consider this article is Quanta Magazine titled Mathematicians Find Long-Sought Building Blocks for Special Polynomials. Figure 7 features a photo of "Benedict Gross (who) was the first to use p-adic numbers to look for the numerical building blocks that Hilbert’s 12th problem asks for. The approach proved successful, decades later."

Figure 7

SageMath can be adapted to help us in changing decimal integers and fractions into p-adic form. Figure 9 shows the conversion of 26353 into its 7-adic form:

Figure 9: permalink

To change a decimal fraction into a so-called "basimal" is not difficult. Using 1/2 and base 7 as an example, here is the SageMath code in blue with output in red:

ring=RealField(30)

ring(1/2).str(base=7)

'0.333333333333'

The 30 just indicates the degree of precision. We see that the 7-ary form of 1/2 is \(0.\overline{3} \). However, this now needs to changed in 7-adic form and to this we need to reverse the order of the digits and place everything to the left of the decimal point. Additionally, 1 must be added to the right-most digit when in the p-adic form:$$0.333333 \dots \rightarrow \dots 333333.0 \rightarrow \dots 333334.0 \rightarrow \overline{3}4.0$$If we multiply this number by 2, the result is 1 and so the representation is correct.

If there is a decimal part in addition to the integer then both parts can be processed together:

n=12.5
n.str(base=7)

'15.333333333333333333'

Changing from 7-ary form to 7-adic form we get: \( \overline{3}4.51 \).