What are p-adic Numbers?

What are the p-adics? They're a specialized class of numbers first uncovered by Kurt Densel in the late 1800s. A key aspect is their absolute value, which depends on the prime number for which any p-adic is based. The p then means the particular prime. Given primes of 2, 3, 5, 7 - for example, one can find 2-adic, 3-adic, 5-adic and 7-adic absolute values which are always computed by taking the reciprocal of the highest multiple of p which divides into any given natural number, N.

If N has no multiples of p, then the absolute value (êp ê)  is simply 1. If we are looking at a p-adic absolute value of zero, the result is always zero. (E.g. [0]_p =ê0 ê_p = 0)

Let's look at some examples before examining more elaborate applications. Consider the 3-adic versions of: 7, 5 and 1/3. What will the 3-adic absolute values be? Compute each in turn:

[7]₃ = 1  (since there are NO multiples of 3 to form the number N = 7)

[5]₃ = 1 (for the same reason, thus: [7]₃ = [5]₃ )

[1/3]₃ = [1]₃/ [3]₃  = 1/ (1/3) = 3

since the reciprocal of 3 is 1/3 which we then divide into the numerator 1.


How about obtaining the p-adic absolute values for each of these: [4]₂  , [1/6]₂ , [1/8]₃ , and [24/25]₂

The first is pretty easy, since: [4]₂ = [2 x 2]₂ = 1/4

The next isn't too terribly difficult either:  [1/6]₂ = [1]₂/ [3 x 2]₂ =
1/(1/2) = 2


and:  [1/8]₃  = [1]₃/ [8]₃ = 1/1 = 1


Note in this case, since the denominator (8) has no 3-factors, it must be that [8]₃ = 1 .

Lastly: [24/25]₂  = [3 x 2 x 2 x 2]₂/ [25]₂ =  (1/8)/ 1 = 1/8

(Again, 25 has no multiples of 2 which can compose it, so [25]₂ = 1)

Even more intriguing are the spatial relations and differences, divergences between normal space and "p-adic" space. Consider the triangle (scalene) shown in Fig. 1 and the linear dimensions (absolute values) of its respective sides. We find: A = 4 ([4 - 0]); B = 6 ([10 - 4]); and C = 10 ([0 - 10]). Now compute the sides using 2-adic absolute values (I will assume the reader can obtain the end computations based on the previous examples):

 

for A: [0 - 4]₂ = 1/4
for B: [4 - 10]₂ = [1/4 - 1/2] = 1/4
for C: [0 - 10]₂ = [10]₂ = 1/2

Amazingly, in the p-adic context we find the counter-intuitive result that side A equals side B (= ¼). In other words, in this context, the triangle is found to be isosceles! A general rubric is that for any such computations of the p-adic absolute values contingent on a given triangle's sides - there will always be found an isosceles triangle - irrespective of how the triangle appears in normal space.

Even more bizarre results await when we examine apparently infinite series in the p-adic context. Thus, a series that first appears to go on to an infinitely large extent may be found much more different when p-adics enter. Consider the series given by the sum:

S = 1 + 5 + (5)² + (5)³ + (5)⁴ + (5)⁵ + (5)⁶ + .........

To treat S p-adically, multiply both sides by 5, then place the result under the original S and subtract:

S = 1 + 5 + (5)² + (5)³ + (5)⁴ + (5)⁵ + (5)⁶ + .........

5S = 5 +   (5)² + (5)³ + (5)⁴ + (5)⁵ + (5)⁶ + .........


Subtract term by term to get: S - 5S = 1 (all other terms above and below cancel out!)

So: -4S = 1 and S = -  ¼

In other words, the sum S is less than 1 in the p-adic venue,  totally counter-intuitive! We see that evidently the notion or concept of "closeness" emerges quite differently - certainly if we can turn an "infinite" (apparently) sum into one yielding a result less than one!

Problems for the Math Maven:

1. For the triangle in Fig. 2, use 7-adic absolute values applied to the sides of the triangle, thereby compute: A, B and C and show it is isosceles.

2. Find the value of the sum S for: 

S = 1 + 7 + (7)² + (7)³ + (7)⁴ + (7)⁵ + ....

3.  Using the series in (2) as written, "invent" a new irrational number based on the p-adic form.