m = d_{0} + 10 d_{1} + 10^{2} d_{2} + ... ,

m = d_{0} + b d_{1} + b^{2} d_{2} + ... ,

In particular, every non-negative integer m has a **2-adic representation**

m = d_{0} + 2 d_{1} + 2^{2} d_{2} + ... ,

The existence of the 2-adic representation of any postive integer
is equivalent to the statement that every positive integer is the sum
of distinct powers of 2 (provided that 1 = 2^{0} is included
as a power of 2).

We shall focus on the 2-adic representation of m.

ord_{2}(m) = the least exponent of 2 in the 2-adic representation of m .

In other words e = ord_{2}(m) if m is divisible by 2^{e}
but not by any higher power of 2. Viewed this way it makes sense to say
that ord_{2}(0) is infinite since 0 is divisible by each power of 2.

We define the **2-adic norm** ||m|| of any integer m by the formula

||m|| = 2^{-e} when e = ord_{2}(m) .

Then we define the **2-adic distance** d(m, n) between two integers
m and n by the predictable formula

d(m, n) = || m - n || .

1 + 2 + 2^{2} + 2^{3} + 2^{4} + ... .

s_{n} = 1 + 2 + 2^{2} + 2^{3} + ... + 2^{n-1}
= 2^{n} - 1 .

To say the least, it is curious that the 2-adic method of summing all of the powers of 2 gives the same result as the method of analytic continuation.

One must bear in mind that the two contexts are completely different. The notion of 2-adic summation rests on the notion of 2-adic limit, which, in turn, rests on the notion of 2-adic distance.

The 2-adic distance d(m, n) has been defined above **only when**
*m* and *n* are integers. Every non-negative
integer has a finite 2-adic expansion, and we have just seen that the
integer -1 has an infinite 2-adic expansion as the sum of all powers
of 2 when infinite sums are interpreted 2-adically.

f = c_{0} + 2 c_{1} + 2^{2} c_{2} + ... ,

ord_{2}(f) = the least j with c_{j} = 1 ,

||f|| = 2^{-e} when e = ord_{2}(f)

d(f, g) = || f - g || .

It is then obvious that any 2-adic integer is the limit, for 2-adic distance, of its sequence of partial sums. Consequently, every 2-adic integer is the limit of a sequence of positive integers.
Moreover, it can be shown that every negative integer is a 2-adic
integer and that every ratio a/b of integers a and b with b **odd**
is a 2-adic integer.

Needless to say, the set of 2-adic integers is not an identifiable subset of the set of real numbers.

But there is a nearly one-to-one correspondence of the set of 2-adic integers with the closed interval [0, 1] of all real numbers between 0 and 1. This correspondence is obtained by associating to the 2-adic integer

f = c_{0} + 2 c_{1} + 2^{2} c_{2} + ...

R(f) = 2^{-1} c_{0} + 2^{-2} c_{1} + 2^{-3} c_{2} + ... .

Further insight into 2-adic analysis may be obtained by pursuing the analogous study of formal power series.

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