How are rational and irrational numbers related?

Question

Autumn Arnold · Accepted Answer

Rational and irrational numbers are mutually exclusive but jointly or collectively exhaustive set of real numbers.

There is only one set of rational and irrational numbers. These two characteristics are crucial. (1) They are mutually exclusive i.e. a number from one set say Rational numbers #QQ# cannot be a member of another set that is irrational numbers #ZZ# and vice-versa. In set theory, we say that the intersection of #QQ# and #ZZ# is #phi#, the null set or #QQnnZZ=phi#. This is because while rational numbers can be written as a ratio of two integers say #p/q#, where #p# and #q# are integers and #q!=0#, irrational numbers cannot be written as such. Irrational numbers in decimal notation have non-terminating non-repeating (or non-recurring decimals), whereas rational numbers terminate after the decimal sign or have non-terminating but repeating (or recurring decimals). (2) However, together they form the set of Real numbers #RR# and both rational and irrational numbers can be represented on real number line and in set theory we say that #QQuuZZ=RR# and there are no real numbers, which do not fall in one or he other category. To put it briefly, a set of real numbers that is jointly or collectively exhaustive but mutually exclusive can be categorized as rational and irrational numbers.

Genesis Amaya · Answer

undefined Rational and irrational numbers are both subsets of real numbers. Rational numbers can be expressed as the quotient of two integers, while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. Together, rational and irrational numbers make up the entire set of real numbers. Additionally, the sum, difference, product, and quotient of a rational number and an irrational number can result in an irrational number. However, the sum, difference, product, or quotient of two rational numbers is always a rational number.