Question Video: Understanding the Sum of Real Numbers Mathematics • Second Year of Preparatory School

Video Transcript

If 𝑎 is a member of the set of integers, 𝑏 is a member of the set of real numbers, and 𝑐 is a member of the set of rational numbers, then 𝑎 plus 𝑏 plus 𝑐 is a member of what?

In this question, we are given three numbers 𝑎, 𝑏, and 𝑐 in three different sets. And we’re asked to determine which set their sum must be a member of. To answer this question, let’s start by recalling what is meant by each of the three given sets of numbers.

First, the set of integers is the set of all whole numbers. Its members include values such as zero, one, two, and so on and negative one, negative two, and so on. Next, we can recall that the set of rational numbers is the set of all quotients of integers with nonzero denominators, that is, all of the numbers that can be written in the form 𝑥 over 𝑦, where 𝑥 and 𝑦 are integers and 𝑦 is nonzero. Finally, we can recall that the set of real numbers is defined to be the union of the set of rational numbers and its complement called the set of irrational numbers. The set includes all numbers that can be written as a quotient of integers and those which cannot.

There are many ways to answer this question. One way is to start by substituting possible values of 𝑎, 𝑏, and 𝑐 into the sum and then analyzing the results. Let’s start with all three values equal to zero. We see that their sum is also zero. This does not give us much information, so let’s try a more complicated example.

To do this, we can recall that there are real numbers that are not rational, such as the square root of two. If we want 𝑏 to be equal to the square root of two, we should choose values for 𝑎 and 𝑏 that allow us to easily analyze the sum of these values. We can keep the values of 𝑎 and 𝑐 at zero so that the sum of the three values is equal to the square root of two.

We can now notice something interesting. Our first sum gave a rational number. However, our second sum gave an irrational number. These are both examples of real numbers, so it appears as though the answer is that the sum must be a real number.

We can now consider why this might be the case. To do this, we need to recall that the addition operation is a closed operation over the set of integers, rational numbers, and real numbers, respectively. These are useful properties to recall since they tell us information about the sets sums of these numbers must be in. For instance, since addition is a closed operation over the set of integers, we know if 𝑥 and 𝑦 are integers, then 𝑥 plus 𝑦 is also an integer.

Similarly, we know that the sum of two rational numbers must be a rational number. We know that both 𝑎 and 𝑐 are rational numbers, since 𝑎 can be written as 𝑎 over one. This means that 𝑎 plus 𝑐 is a rational number.

We can use this idea with the closure of addition in the real numbers and the commutativity and associativity of addition to consider the sum of 𝑎, 𝑏, and 𝑐. We know that all rational numbers are real numbers, so 𝑎 plus 𝑐 and 𝑏 are real numbers. The closure property of the addition of real numbers then tells us that the sum of these numbers is a real number. We can then use the commutativity and associativity of the addition of real numbers to rewrite this as 𝑎 plus 𝑏 plus 𝑐. This means the sum of these numbers must be a real number because each term itself is real.

Finally, it is worth noting that we cannot choose a strict subset of the set of real numbers for our answer. To see this, we can note that we can choose 𝑎 equals zero and 𝑐 equals zero to get that 𝑎 plus 𝑏 plus 𝑐 equals 𝑏. This means that we can choose any real number 𝑏 and the sum will give 𝑏. Hence, this sum can be equal to any real number. This means that the answer must be that the sum of these three numbers is an element of the set of real numbers.