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A Path through the Lattice Points in a Quadrant

step
120
Let
+
={1,2,3,}
be the set of positive integers. The set of lattice points in the first quadrant is the set
+
×
+
={(1,1),(1,2),}
, where both coordinates are positive integers. Even though
+
×
+
is two-dimensional, it is possible to set up a one-to-one correspondence between
+
×
+
and
+
, as shown in the picture.
By associating
m
n
with the lattice point
(m,n)
, the path through the lattice points gives an enumeration of the positive unreduced rational numbers. Skipping past the fractions that have a common factor gives a listing of the positive rational numbers. The matching shows that there are as many positive fractions as positive integers.
In spite of that, there are differences between the integers and the rationals; for example, between any two rationals there is another rational.
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