The fraction spiral
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One way to replace a fraction with an integer is to represent each fraction n/d by the point with coordinates (n, d), then traverse all the points with integer coordinates using a spiral like the one shown opposite. For example, the fraction 3/2 (corresponding to the point (3, 2)) is assigned the number 29.
This is a standard method for proving that the rational numbers are countable (in other words, that "there are no more fractions than integers"). It can, however, be improved, since different fractions may represent the same number (for example, 3/2 = 6/4 = (-9)/(-6), etc
Joining the numbers
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