Example (h) illustrates a technique for showing that any repeating decimal number is ratio...

Example (h) illustrates a technique for showing that any repeating decimal number is rational. A calculator display shows the result of a certain calculation as 40.72727272727. Can you be sure that the result of the calculation is a rational number? Explain.

Example

Determining Whether Numbers Are Rational or Irrational

a. Is 10/3 a rational number?

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b. Is −  a rational number?

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c. Is 0.281 a rational number?

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d. Is 7 a rational number?

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e. Is 0 a rational number?

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f. Is 2/0 a rational number?

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g. Is 2/0 an irrational number?

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h. Is 0.12121212 …a rational number (where the digits 12 are assumed to repeat forever)?

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i. If m and n are integers and neither m nor n is zero, is (m + n)/mn a rational number?

Solution

a. Yes, 10/3 is a quotient of the integers 10 and 3 and hence is rational.

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b. Yes, −  = − which is a quotient of the integers −5 and 39 and hence is rational.

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c. Yes, 0.281 = 281/1000. Note that the real numbers represented on a typical calculator display are all finite decimals. An explanation similar to the one in this example shows that any such number is rational. It follows that a calculator with such a display can represent only rational numbers.

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d. Yes, 7 = 7/1.

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e. Yes, 0 = 0/1.

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f. No, 2/0 is not a number (division by 0 is not allowed).

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g. No, because every irrational number is a number, and 2/0 is not a number. We discuss additional techniques for determining whether numbers are irrational in Sections.

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h. Yes. Let x = 0.12121212 …Then 100x = 12.12121212 …Thus

100x − x = 12.12121212 …− 0.12121212 …= 12.

But also 100x − x = 99x by basic algebra

Hence 99x = 12,

and so 

Therefore, 0.12121212 …= 12/99, which is a ratio of two nonzero integers and thus is a rational number.

Note that you can use an argument similar to this one to show that any repeating decimal is a rational number. In Section 9.4 we show that any rational number can be written as a repeating or terminating decimal.

i. Yes, since m and n are integers, so are m + n and mn (because sums and products of integers are integers). Also mn ≠ 0 by the zero product property. One version of this property says the following:

Zero Product Property

If neither of two real numbers is zero, then their product is also not zero.

(See Theorem and exercise 1 at the end of this section.) It follows that (m + n)/mn is a quotient of two integers with a nonzero denominator and hence is a rational number.

Theorem

T11. Zero Product Property If ab = 0, then a = 0 or b = 0.

Exercise 1

Determine which statement are true and which are false. Prove those that are true and disprove those that are false.

Exercise

The difference of any two irrational numbers is irrational.