Rounding p-adic Rationals

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Rational -adic numbers share the same decimal notation and math as do repeating decimals. A vinculum (or overbar) is also allowed in -adic number notation to indicate repeated digits. However, a nonzero vinculum must appear only on the leftmost digits, while normal repeating decimals use a vinculum only on the right. This Demonstration rounds to only repeating digits.

Contributed by: Robert L. Brown (December 2013)
Open content licensed under CC BY-NC-SA



In this Demonstration, -adic rounding is accomplished by truncating repeating digits of on the left. This is equivalent to replacing the integer part of with , where is the floor function.

There is a one-to-one mapping between the two notations for the rational numbers, but in what sense can truncation be considered rounding for -adic numbers? Multiplying the -adic rational notation by the value of the denominator we see something like the numerator. This is a useful motivation for students being introduced to the -adic norm.

The in -adic stands for prime. A -adic system is defined by which prime is chosen to be . This Demonstration allows "10-adic" numbers. Although 10-adic is not a proper -adic system, it does behave like a -adic representation of the rational numbers. The 10-adic option is included because students are familiar with base 10 numbers.

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