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Task/P-Adic-numbers-basic/00-TASK.txt

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+;Conversion and addition of [[wp:P-adic_number|p-adic Numbers]].
+
+
+;Task.
+
+Convert two rationals to p-adic numbers and add them up.
+Rational reconstruction is needed to interpret the result.
+
+p-Adic numbers were introduced around 1900 by Hensel. p-Adic expansions
+(a series of digits 0 ≤ d < p times p-power weights)
+are finite-tailed and tend to zero in the direction of higher positive
+powers of p (to the left in the notation used here).
+For example, the number 4 (100.0) has ''smaller'' 2-adic norm than 1/4 (0.01).
+
+If we convert a natural number, the familiar p-ary expansion is obtained:
+10 decimal is 1010 both binary and 2-adic. To convert a rational number a/b
+we perform p-adic long division. If p is actually prime, this is always possible
+if first the 'p-part' is removed from b (and the p-adic point shifted accordingly).
+The inverse of b modulo p is then used in the conversion.
+
+'''Recipe:''' at each step the most significant digit of the partial remainder
+(initially a) is zeroed by subtracting a proper multiple of the divisor b.
+Shift out the zero digit (divide by p) and repeat until the remainder is zero
+or the precision limit is reached. Because p-adic division starts from the right,
+the 'proper multiplier' is simply
+d = partial remainder * 1/b (mod p).
+The d's are the successive p-adic digits to find.
+
+Addition proceeds as usual, with carry from the right to the leftmost term,
+where it has least magnitude and just drops off. We can work with approximate rationals
+and obtain exact results. The routine for rational reconstruction demonstrates this:
+repeatedly add a p-adic to itself (keeping count to determine the denominator),
+until an integer is reached (the numerator then equals the weighted digit sum).
+But even p-adic arithmetic fails if the precision is too low. The examples mostly
+set the shortest prime-exponent combinations that allow valid reconstruction.
+
+
+;Related task.
+
+[[p-Adic square roots]]
+
+
+;Reference.
+
+[https://www.cut-the-knot.org/blue/p-adicExpansion.shtml] p-Adic expansions
+
+
+__TOC__
+
+