### Video Transcript

What is the union of ℚ and ℚ
prime?

In this question, we are asked to
describe the union of two sets. The easiest way to answer this
question is to recall the result that ℚ is the set of rational numbers and ℚ prime
is the set of irrational numbers, and their union is the set of real numbers. While this is enough to answer this
question, it is always worth understanding where these results come from and why
they are defined in this way. So, it is often worth looking at
why these definitions are chosen.

To explore this, let’s start by
recalling the definition of the set of rational and irrational numbers. First, the set of rational numbers
is the set of all quotients of integers such that the denominator is nonzero. We can also recall that the set of
rational numbers is also the set of all numbers with a finite or repeating decimal
expansion. We have also seen that numbers such
as 𝜋 or the square root of two do not have a finite or repeating decimal
expansion. In other words, their decimal
expansion is infinite and it does not repeat. This means that the set of rational
numbers does not include all numbers with decimal expansions.

We can write the set of nonrational
numbers by taking the complement of the set of rational numbers. This is the set of irrational
numbers. We can take the union of these two
sets to construct a set containing all numbers with a finite or repeating decimal
expansion and all numbers with an infinite decimal expansion. We call this the set of real
numbers ℝ. This means that the set of real
numbers is really just the set of all numbers we can represent using a decimal
expansion. And it is constructed as the union
of the disjoint set of rational numbers and irrational numbers.