Free Math Worksheets

- What distinguishes a repeating decimal from a nonrepeating decimal? - To convert a repeating block of two digits, multiply by \(100\) and subtract the original number. - A pattern governed by a rule is not necessarily periodic.

1. The decimal for \(x\) repeats: \(x = 1.\overline{23}\). Therefore, \(x\) is rational. 2. The decimal for \(y\) is nonterminating and nonrepeating because the number of consecutive \(2\)s keeps increasing. Therefore, \(y\) is irrational. 3. Let \(x = 1.\overline{23}\). Then \(100x = 123.\overline{23}\). Subtracting gives \(99x = 122\), so \(x = \frac{122}{99}\). The numerator and denominator have no common factor, so the fraction is in lowest terms.

1. \(x\) is rational, and \(y\) is irrational. 2. The decimal for \(x\) repeats, while the decimal for \(y\) is nonterminating and nonrepeating. 3. \(x = \frac{122}{99}\)

- Convert a repeating two-digit block by multiplying by \(100\) and subtracting. - Compare decimals digit by digit from left to right. - Every terminating decimal is rational.

1. Since \(x = 0.121212\ldots\), \(x = \frac{12}{99} = \frac{4}{33}\). 2. The decimal for \(y\) contains blocks of \(1\)s and \(2\)s whose lengths keep increasing. Therefore, no fixed block repeats forever. The decimal is nonterminating and nonrepeating, so \(y\) is irrational. 3. Compare \(x = 0.121212\ldots\) with \(y = 0.121122\ldots\). At the fourth decimal place, \(x\) has \(2\) and \(y\) has \(1\), so \(y < x\). 4. One rational number between them is \(z = 0.1212\). Indeed, \(0.121122\ldots < 0.121200\ldots < 0.121212\ldots\).

a) \(x = \frac{4}{33}\) b) \(y\) is irrational because its increasing block lengths make its decimal expansion nonterminating and nonrepeating. c) One example is \(z = 0.1212\), since \(y < 0.1212 < x\).