Rational Number

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 (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:

\mathbb{Z} = \left{\frac{a}{b} \mbox{ for all } a,b \in \mathbb{Z}, b\ne0\right}

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

\frac{\pi}{1}

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:

\begin{array}{rclcl}7/3&=&2,33333\ldots&=&2,\overline{3}\\16/11&=&1,454545\ldots&=&1,\overline{45}\\11/6&=&1,833333\ldots&=&1,8\overline{3}\end{array}

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?
Divided by zero