The decimal (0.333\ldots) (or (0.\overline3)) is generated by the fraction (\frac13). Therefore, (\frac13) is the geratrix fraction of (0.\overline3). 2. Types of Decimals | Type | Description | Example | |------|-------------|---------| | Terminating decimal | Ends after finite digits | (0.25 = \frac14) | | Pure repeating decimal | All digits after the decimal point repeat | (0.\overline142857 = \frac17) | | Mixed repeating decimal | Some non-repeating digits followed by a repeating block | (0.1\overline6 = \frac16) |
(0.3\overline18) (x = 0.3181818\ldots) Multiply by 10: (10x = 3.181818\ldots) (now pure repeating: (3.\overline18)) (1000x = 318.181818\ldots) (since (10x \times 100 = 1000x)) Wait — better method: Let (x = 0.3\overline18) Multiply by 10: (10x = 3.\overline18) (pure repeating) Now (10x = 3 + 0.\overline18) (0.\overline18 = \frac1899 = \frac211) So (10x = 3 + \frac211 = \frac33+211 = \frac3511) Thus (x = \frac35110 = \frac722) exercicios sobre fração geratriz
(0.375 = \frac3751000 = \frac38) Case 2: Pure Repeating Decimal Let (x) be the decimal. Multiply by (10^n) where (n) is the length of the repeating block. Subtract the original equation to eliminate the repeating part. The decimal (0
(0.\overlineabc\ldots = \frac\textrepeating block10^n - 1) where (n) = number of digits in the block. Types of Decimals | Type | Description |
(0.\overline72) (x = 0.727272\ldots) (100x = 72.727272\ldots) Subtract: (100x - x = 72 \Rightarrow 99x = 72 \Rightarrow x = \frac7299 = \frac811) Case 3: Mixed Repeating Decimal Let (x) be the decimal. Multiply by a power of 10 to move the decimal point to just before the repeating block, and by another power to include the whole repeating part. Subtract.
Открытый доступ