Allan is ordering a set of rational numbers that includes positive values, negative values, fractions, and decimal numbers. How can he order them?

Rational numbers with the same denominator are ordered by the value of their numerator. Rational numbers with different denominators can often be compared to some known value: > 1/2 or < 1/2, for example.

Decimal numbers are ordered by the place values of their digits. In general, all numbers can be converted to a decimal representation with sufficient precision to permit ordering.

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Algorithms

There are various algorithms available for ordering. Allan can make use of any of these. Some are best suited to use by computer programs, while others can be used for manual ordering.

For example, Allan could plot the numbers one at a time on a number line. This is similar to the method "insertion sort" where a number is placed in a list relative to other numbers already placed.

Allan could also order small groups of numbers, then merge the groups into larger groups. Merging two already-ordered lists is generally easier than ordering one large list. The list size can be as large or small as convenient, including as few as 2 numbers at a time.

Of course, decimal numbers are ordered by comparing digits left-to-right. Two numbers with most-significant digits having different place values will be ordered by the place value of the digits. Numbers with digits having the same place value will be ordered by the value of the digit. Numbers with same-digits will be ordered by the next least-significant digits.