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Bot GPT 3.5 A repeating decimal is a decimal number in which one or more digits repeat infinitely after the decimal point. For example, the decimal representation of the fraction 1/3 is 0.333..., where the digit 3 repeats infinitely.
To convert a repeating decimal to a fraction, you can follow a specific process. Let's denote the repeating decimal as "0.abcd..xyzxyzxyz..." where the digits "abcd...xyz" repeat infinitely.
1. Let's assume there are "n" repeating digits.
2. Multiply both sides of the repeating decimal by a power of 10 to remove the repeating part.
In our example, multiply by 10^n to eliminate the repeating part.
This gives us: 10^n * (0.abcd..xyzxyzxyz...) = abcd..xyz.xyz.xyz... (1)
3. Subtract equation (1) from the original repeating decimal.
This gives us: (0.abcd..xyzxyzxyz...) - abcd..xyz.xyz.xyz... = (0.xyz.xyz.xyz...) (2)
4. In equation (2), the number of non-repeating digits (xyz) is equal to the value of the repeating decimal without the repeating part.
5. Let's assume that equation (2) equals "x" (the non-repeating part without the repeating part).
6. To convert the repeating decimal to a fraction, divide "x" by a number consisting of the same number of nines as the repeating part has digits.
In our example, divide "x" by 999... (n nines).
This gives us: x / 999... = fraction form of the repeating decimal.
So, the repeating decimal can be represented as a fraction by dividing the non-repeating part by a number consisting of the same number of nines as the repeating part has digits.