First uncovered by Kurt Densel in the late 1800s, the p-adics are a specialized class of number whose key aspect is their absolute value. This depends on the prime number for which any given 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 ) 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]

[1/3]

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:

[5]

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

_{3}= [1]_{3}/ [3]_{3}= 1/ (1/3) = 3since 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]

The first is pretty easy, since: [4]

The next isn't too terribly difficult either:

_{2}, [1/6]_{2}, [1/8]_{3}, and [24/25]_{2}The first is pretty easy, since: [4]

_{2}= [2 x 2]_{2}= 1/4The next isn't too terribly difficult either:

[1/6]

_{2}= [1]_{2}/ [3 x 2]_{2 }= 1/(1/2) = 2*And*: [1/8]

In this case, since the denominator (8)

Lastly: [24/25]

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

for B: [4 - 10]

for C: [0 - 10]

Amazingly, in the p-adic context we find the counter-intuitive result that side

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)

To treat S p-adically, multiply both sides by

5S = 5 + (5)

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

So: -4S = 1 and S = -

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!

1. For the triangle shown in Fig. 2 below:

_{3 }= [1]_{3}/ [8]_{3}= 1/1 = 1In this case, since the denominator (8)

*has no 3-factors*, it must follow 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 below:

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}= 1/4for B: [4 - 10]

_{2}= [1/4 - 1/2] = 1/4for C: [0 - 10]

_{2}= [10]_{2}= 1/2Amazingly, 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

**, then place the result under the original S and subtract, e.g.:***5*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!

*:***Suggested Problems**1. For the triangle shown in Fig. 2 below:

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:

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

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.

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