Real number system

Question

Is the set of rationals a subset of the irrationals? I always assumed it was, but given that irrationals are defined to be numbers that have an infinite, non-repeating decimal expansion, there cannot possibly exist a number in both sets?

Answer 1

That's not the definition of irrational numbers.

A rational real number is a number that can be written as the ratio of two integers, $\displaystyle \frac{p}{q}$.

An irrational real number is a real number that is not rational.

It turns out that this will imply an irrational number has a decimal expansion which is not a repeated sequence of digits. Consequently, every rational real number that does have an infinite decimal expansion does consist of a repeated pattern, for example $\displaystyle \frac{1}{7} = 0.142857142857...$

Given these new definitions of rational and irrational, it's a good exercise to prove this!

Answer 2

Rational number is the form of p/q where p and q are integer and prime to each other. Other than Rational number the rest of real rumber is said to be irrational number.