1. What is a Geratrix Fraction? A geratrix fraction (Portuguese: fração geratriz ) is the common fraction that generates a repeating decimal (also called a recurring decimal). In other words, it is the fraction in lowest terms that, when divided, produces a given decimal expansion that eventually repeats.
(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.
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.\overlineabc\ldots = \frac\textrepeating block10^n - 1) where (n) = number of digits in the block.
(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)
Both terminating and repeating decimals have a geratrix fraction. Irrational decimals (e.g., (0.1010010001\ldots)) do not. Case 1: Terminating Decimal Write the decimal as a fraction with a power of 10 in the denominator, then simplify.