What numbers are always rational?

Answer 1

Any given number is either rational or it is not.

A rational number can be expressed as a fraction (where the denominator (the number at the bottom of the fraction) is not 0). This really is all there is to it.

So the answer to your question is any number that can be expressed as a fraction where the denominator is not 0.

Examples:

The number 1 is rational (as are all integers) as it can be written as 1/1.

The number 1.656 is rational as it can be written as 1656/1000.

The number 0.3 recurring (i.e. there are an infinite number of 3's after the decimal point) is rational because it can be written as 1/3. Although the digits continue infinitely there is a recurring pattern and so it is rational.

The only numbers that are irrational are those that are made up of an infinite series of digits AND do not have a recurring pattern.

Examples:

Pi (the number that represents the ratio between the circumference of a circle and its diameter) is irrational. It can be approximated to 22/7 but this is only an approximation. The real value has an infinite number of digits (which do not repeat in any pattern - they are seemingly completely random) after the decimal point. It can't be translated in to a fraction for this very reason.

The square root of 2 is also irrational. It is 1.4... where ... indicates that an infinite number of digits follow. There is no recurring pattern to the series of digits.