Rational number
The cardinality of the rational numbers is 0א.
Explanation
You can put all the rational fractions in a table.
1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 ⋯ 1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 ⋯ 1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3 9/3 ⋯ 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 ⋯ 1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 ⋯ 1/6 2/6 3/6 4/6 5/6 6/6 7/6 8/6 9/6 ⋯ 1/7 2/7 3/7 4/7 5/7 6/7 7/7 8/7 9/7 ⋯ 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 ⋯ 1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 9/9 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
If you go diagonally through the table, you can count these fractions
1/1 2/1 3/1 4/1 5/1 1/2 2/2 3/2 4/2 1/3 2/3 3/3 1/4 2/4 1/5
This shows that the cardinality of rational numbers is countably infinite and is equal to the cardinality of the natural numbers and is = the set of integers = the set of algebraic numbers.
HistoryThe one-to-one correspondence was introduced by the German mathematician Georg Cantor (1845 - 1918). |