| What Does Ardee Do? |
Ardees is a specialized calculator that only performs long division. Why only long division? Most computers and calculators designed for science and engineering typically perform numerical operations that are accurate out to 15-17 places to the right of the decimal, at which point they are rounded off and truncated. Many fractions can be carried out much further than this, but for most purposes such accuracy is unnecessary. Unfortunately such accuracy IS necessary to study the patterns found in repeating decimals. Since normal machine computations can't obtain these long fractions, Ardee resorts to old-fashioned but very precise long division.
Ardee performs long division carried out to any number of places to the right of the decimal. (I say "any number" but there are limits to what Ardee can display; more on this below.) When a fraction is obtained, Ardee checks to see if a repeating pattern emerges. When Ardee discovers a repeating pattern, it captures the repeater and the point in the fraction where it starts, then moves onto the next fraction. Ardee is thus designed to display a whole series of fractions, so that patterns may be seen within the series as well as within each fraction. Such series are defined by a range of dividends and a divisor: 1/N through n/N. Ardee also looks at such series and displays how they cluster according to the period of the repeater and where it starts in the fraction. |
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| Terminology |
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Ardee's terminology is adopted from the language of mathematics:
Dividend: the number to be divided
Divisor: the number to divide with
Quotient: the result of the division
Remainder: the portion of a quotient that's "left over" when the divisor doesn't go evenly into the dividend
Reptend: the portion of the remainder that repeats endlessly.
Period: the length of a reptend
Ardee's main interest is reptends. |
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| Limitations |
Calculations using very large number sets may not be displayable in Ardee. For instance, the divisor 117,649 has fractions with repeating periods of up to 100,943. To apply the divisor to the entire range of 1 through 117,649 would require 12 GB just to display the fraction list, which is beyond what Ardee can display. In addition, fractions like these require long division to be carried out to tens of thousands or even hundreds of thousands of places. This can take a long time for a big series of fractions. To mitigate this somewhat, Ardee's output can be redirected to delimited files which have no limit beyond the available disk space on your computer. Alternatively, such huge fraction series can be broken up in to small consecutive chunks that CAN be displayed one at a time.
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