0
Consider the sequence (a_i) where a_i is pi rounded to the i_th decimal place. This sequence clearly contains only rational numbers since every number in it has a finite decimal expansion. Furthermore this sequence is Cauchy since a_i and a_j can differ at most by 10^(-min(i,j)) or something which can be made arbitrarily small by choosing a lower bound for i and j. Now note that this sequence converges to pi in the reals, so it can not converge in the set of rational numbers. Therefore the rational numbers allow a non-convergent Cauchy sequence and are thus by definition not complete.
Still curious? Ask our experts.
Chat with our AI personalities
Jordan
Blake
ReneAdd your answer:
Earn +20 pts
Is the intersection of the set of rational numbers and the set of whole numbers is the set of rational numbers?
No, it is not.
Are natural numbers the same of rational numbers?
The set of rational numbers includes the set of natural numbers but they are not the same. All natural numbers are rational, not all rational numbers are natural.
A set of numbers combining rational and irrational numbers?
The Real numbers
What is set of rational numbers union with integers?
It is the rational numbers.
Does a real number contain the set of rational numbers?
No. A real number is only one number whereas the set of rational numbers has infinitely many numbers. However, the set of real numbers does contain the set of rational numbers.
