Rational numbers are all numbers that can be expressed as division of two integers. In other words rational numbers are *ratio* of two integers. We denote the set of rational numbers as *ℚ* (for “quotient”). For example, 1, 42, 1.5, -3/7 are rational numbers; *π* is not rational number since it cannot be expressed as division of two integers. We can define this set as follows:

Be aware of the condition *a, b ∈ ℤ*. It’s tempting to write

and claim that *π* is a rational number. It’s not because the fraction is not a valid fraction according to our definition. The numerator is not an integer thus it’s not a rational number even though it’s a fraction.

We can define a rational number another way: it is a number which decimal representation is either finite or repeating.

*5*is a rational number since it’s decimal representation is finite.*− 47.42*is a rational number since it’s decimal representation is finite.*3.666666…*is a rational number. It’s decimal representation is not finite, it’s infinite, but it’s repeating and the number can be written as a fraction. In this case it’s*11/3 = 3.6666…**π*is not rational number since the decimal representation is infinite and is not repeating. Another example is*√2*or Euler number.

You can see two notations of repeating numbers. The first one is the “…” symbol and the second one is a line above the repeating sequence. See examples:

## Properties of Rational Numbers

- Since rational numbers contains the integers, the set of all rational numbers is infinite set.
- Rational numbers are closed under addition. It means that if you add any two rational numbers you will always get another rational number. Try it! E. g.
*5.5 + 4.5 = 10*or*− 6.2 + 3.8 = − 2.4*. All of them are rational numbers. - The same goes for multiplication. We can multiply any two rational numbers and we always get another rational number. E. g.
*2.2 · 7.7 = 16.94*. - The same goes for subtraction. E. g.
*7.1 − 6.9 = 0.2*, all of them rational numbers. - Rational number are
*almost*closed under division. We cannot divide by zero. Thus if we take*a = 5*and*b = 0*, both rational numbers, we cannot compute*a/b*since*5/0*is not defined. As a corollary rational numbers are not closed under division. But they are closed*except*for division by zero. So just please… Don’t divide by zero. You know what will happen, right?