Use the methods in Exercises 1–7 to show that both .74999 ⋯ and .75000 ⋯ are decimal expansions of 3/4. [Every terminating decimal can also be expressed as a decimal ending in repeated 9’s. It can be proved that these are the only real numbers with more than one decimal expansion.]
In Exercises 1?7, express the given repeating decimal as a fraction.
Exercises 1?7
1.
.373737 ⋯
2.
.929292 ⋯
3.
76.63424242 ⋯ [Hint: Consider 10,000d − 100d, where d = 76.63424242 ⋯.]
4.
.13513513 ⋯ [Hint: Consider 1000d − d, where d = 13.513513 ⋯.]
5.
.135135135 ⋯ [Hint: Consider 1000d − d, where d = 13.513513 ⋯.]
6.
.33030303 ⋯
7
52.31272727 ⋯
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