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Subsection 6.2.4 Rational and Irrational Numbers

Let’s define two more useful classes of numbers.

Definition of Rational Numbers

A rational number is a number that can be expressed as the quotient of two integers. So, for example, the following are all rational numbers:

5, \(\frac{7}{11}\text{,}\) \(\frac{3475}{58}\text{,}\) \(\frac{- 7}{985}\)

In case you’re thinking that 5 doesn’t appear to be a quotient, notice that we could have written it as \(\frac{5}{1}\text{.}\)

We’ll leave writing this definition as a logical expression as a problem for you to solve.

Definition of Irrational Numbers

An irrational number is a real number that isn’t rational. Two important irrational numbers are π and \(\sqrt{2}\text{.}\) There are, of course, infinitely many more.

Big Idea

Modern mathematics could not exist without the logical tools that we have been describing.

Exercises Exercises

1.

Which (one or more) of the following proposed definitions for the rational numbers is correct:

I. ∃x, y, z (Rational(x) ≡ (x = \(\frac{y}{z}\)))

II. ∀x (Rational(x) ≡ (∃y, z (x = \(\frac{y}{z}\))))

III. ∃x (∀y, z (Rational(x) ≡ (x = \(\frac{y}{z}\))))

  1. Just I.

  2. Just II.

  3. Just III.

  4. Two of them

  5. All three of them

Answer.
Correct answer is B.
Solution.
Explanation: II is correct. To see why I isn’t, consider: Let x be some nonrational number (for example, ). Then Rational(x) is false. Now let y and z be 1. Then (x = y/z) is also false. So the two expressions are equivalent. Thus I is trivially true. The required values of x, y, and z exist. But it doesn’t tell us anything about what numbers are or are not rational. The problem with III is similar. Let x be some nonrational number, such as . So Rational(x) is false. y, z ( = y/z) is also false. The two expressions are equivalent. The claim is true. But, again, we have no way to decide whether or not a given number is rational.