Proving That a Set Is Closed

Instructor: Christopher Haines
In this lesson, we prove the set of rational numbers is closed under the operation of addition. Following the proof, we deduce that a number having a terminating decimal representation is rational.

Rational and Irrational Numbers

A rational number is any number that can be expressed in the form of a quotient of two integers (a fraction). So, r is rational if


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We know that m and n are integers and n ≠ 0. Examples of rational numbers include 3/4, 2/3, 1000/10018, 0, -2 and -7/11. If expressing a number r in this way is not possible, we say the number r is irrational. Some examples of irrational numbers include:


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Adding/Subtracting Two Rational Numbers

There is a systematic method of adding two rational numbers. Suppose we would like to add 3/4 and 2/3. To begin, we list the multiples of the denominators, 3 and 4, respectively.


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and


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Note that 12 appears in both of these rows, and it is also the smallest common value. Therefore, 12 is the least common multiple of 3 and 4. Next, we can convert 3/4 to a fraction with the denominator of 12 by multiplying the numerator and denominator by 3 as follows.


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What we are really doing is multiplying 3/4 by 3/3 = 1. Similarly, by multiplying the numerator and denominator of 2/3 by 4 we obtain:


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Now that both denominators are the same, we can proceed to add 3/4 to 2/3 to obtain


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Although using the least common multiple of the two denominators succeeds when adding two rational numbers, it is not necessary that the common multiple be the least common multiple. For instance, consider the arithmetic:


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Note that 16 multiplied by 7 is 112, and therefore 112 is a multiple of 16 and 7 (a common multiple). We can then rewrite the problem as:


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This simplifies to:


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Proof of Closure Under Addition

Now that we understand how to add two rational numbers, we can show that the rational numbers are closed under addition. By closed under addition, we mean that if r and s are rational numbers, then r + s is also a rational number. In the last section, we used the method of finding the least common multiple of the two denominators, although it would suffice to simply find a common multiple.

When proving such a general statement as this, it is not enough to take fractions such as 7/8 and -9/13, add the two numbers and show that the end result is a rational number. We need to use arbitrary rational numbers instead, and that amounts to introducing letters rather than numbers. So, suppose r and s are rational numbers and:


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and


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where m, n, o and p are all integers, and n and p are both nonzero. A common multiple of n and p is the product of the two, np. Converting the r and s into fractions both having np as the denominator, we see that:


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Simplifying the far right side of this equation, we obtain:


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Since the product of two integers is an integer, mp, on and np are all integers. Further, since the sum of two integers is again an integer, mp + on is also an integer. Finally, since mp + on and np are both integers, r + s is a rational number (by definition). We have now proven the claim that the rational numbers are closed under addition.

Terminating Decimal Representations

Consider the number 0.7568. This is an example of a number with a terminating decimal representation because from the fifth decimal place on, there are only zeroes. We can write this number as:


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The right side of this equation consists of the sum of four rational numbers. Since we know that the sum of two rational numbers is a rational number, it follows that the sum of four rational numbers is a rational number. Therefore, 0.7568 is a rational number. More generally, any number with a terminating decimal sequence can be shown to be rational using this exact same argument.

Non-Terminating Decimal Representations

The decimal representation for a number need not terminate. For instance, a standard calculator might indicate that:


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