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Question Video: Finding the Union of Number Sets and Their Complements Mathematics • 8th Grade

What is ℚ ∪ ℚ′?

02:06

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.

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