A non-terminating but recurring decimal is

Every real number is either rational or irrational, based on its standard decimal representation. A rational number is any number that can be written in the form p/q (q is not equal to 0); its standard decimal form either terminates after a finite number of digits, or is non-terminating but recurring, meaning a fixed block of digits repeats forever. An irrational number's decimal expansion is non-terminating and non-recurring - it never settles into a repeating pattern.

Here the decimal is described as non-terminating but recurring - it goes on forever, but with a repeating block. That is precisely the second case in the definition of a rational number: for example, 0.777... (7 recurring) equals 7/9, and 0.181818... equals 2/11 - both are non-terminating, recurring decimals that reduce to a p/q fraction.

• An integer, such as 5, or a natural number, such as 5, is normally written with a decimal expansion that terminates (5 = 5.0); it is not the kind of number that a non-terminating recurring decimal is used to represent, so neither fits here.

• An irrational number, such as the square root of 2 (1.41421356...) or pi (3.14159265...), is non-terminating and non-recurring - no block of digits ever repeats - which is the opposite of what is described here.

So a non-terminating but recurring decimal can always be expressed as p/q, which makes it a rational number.