For example, fraction 1478⁄517 is in the 1478th row and 517th column. Every fraction is somewhere in the table. You should be able to see the pattern: in the first row, we have always fractions in the form 1⁄q, on the second row 2⁄q while in the first column we have fractions in the form p⁄1, in the second row p⁄2, etc. We can prove there is the same amount of rational numbers as there are natural numbers.īut how can we pair each rational number to an unique natural number? Imagine all rational numbers written in the form table as follows: Hence one would say there must be more rational numbers than natural numbers, right? I can sense your spider-sense is tingling! Your intuition is correct. There is even an infinite amount of rational numbers between numbers 0.01 and 0.02. On the other hand, we have an infinite amount of rational numbers between the numbers 2 and 4. We have only one integer between numbers 2 and 4 and that is number 3. Be aware that every integer is also a rational number. Anything that can be denoted as a fraction p⁄q where p and q are integers is a rational number. Comparing Natural Number and Rational Numbers The set of natural numbers contains all the even numbers and moreover, it contains another infinite amount of odd numbers. It means… The set of the natural numbers has the same size as the set of all even numbers! And each even number can be paired to some unique natural number. Suddenly, each natural number can be paired to some unique even number.
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