Rational numbers are countable, so an order can be placed on them just like the natural numbers. Although such an ordering is not obvious (nor unique), one such ordering
can be achieved by letting the -axis represent the denominator of a rational number and the
-axis it numerator. Then starting at the
origin, spiral clockwise away from it, so that each integer pair represents a rational
number. Each time that the -axis is passed, the denominator will be 0, which generates
an illegal number, and all such numbers need to be skipped. Repeats, such as and , also occur, so only the first occurrence of a rational
number should be included in the spiral. Finally, since numbers like can be reduced, such repeats must also be skipped.

The pairs of numbers so obtained are 0/0, 1/0, 1/1, 0/1, , , , , , , ..., giving the points (0, 0), (1, 0), (1, 1), (0, 1),
, , , , , , ....