What are the

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

**p-adics**? They're actually 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 on 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 is simply 1. If we are looking at a p-adic absolute value of zero, the result is always zero. (E.g. [0]

_{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}

[24

**/**25 ]

_{2}

The first is pretty easy, since: [4]

_{2 }= [2 x 2]

_{2}= 1 /4

The next isn't terribly difficult either:

[1

**/**6 ]_{2 }= [1 ]_{2}**/**[3 x 2 ]_{2}= 1/(1/2) = 2and: [1

**/**8 ]

_{3}= [1 ]

_{3}

**/**[8 ]

_{3}= 1/ 1 = 1

Note in this case, since the denominator (8)

**t must be that [8 ]**

*has no 3-factors*, i_{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 perform 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*. 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 a 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}+ .+ .........

5 S = 5 + (5)

^{2}+ (5)

^{3}+ (5)

^{4}+ (5)

^{5}+ (5)

^{6}+ .+ .........

______________________________________________

S - 5S = 1 (all other terms above and below cancel out!)

So:

In other words, the sum S

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)

S - 5S = 1 (all other terms above and below cancel out!)

So:

**-4S = 1**and**S = -1/4**In other words, the sum S

**is less than 1**in the p-adic form - 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 whiz:__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 (2) as written, "invent" a new irrational number based on the p-adic form.

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