Reciprocals and repeating digits

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In summary, reciprocals are numbers that result in a product of 1 when multiplied together. To find the reciprocal of a number, divide 1 by that number. Repeating digits are numbers with a repeating pattern after the decimal point, and are represented using a bar over the pattern in mathematical notation. To convert a repeating decimal to a fraction, the decimal represents a fraction with the repeating pattern as the numerator and the number of repeating digits as the denominator.
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guysensei1
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So when you take the reciprocals of integers, some have nice non repeating decimal forms like 0.5 and 0.2, but of course a lot of them give infinite repeating strings.

1/7 gives 0.142857 repeating, so there are 6 digits repeating.
1/12 give 0.083333... but there is only 1 digits that's repeating, which is 3.

Which integer reciprocal gives the most digits repeating? Or is there no answer?
 
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  • #2
Consider the sequence ##\frac19, \frac{1}{99}, \frac{1}{999}, \frac{1}{9999},\ldots##
 

FAQ: Reciprocals and repeating digits

What are reciprocals?

Reciprocals are numbers that, when multiplied together, result in a product of 1. For example, the reciprocal of 2 is 1/2, and the reciprocal of 3 is 1/3.

How do you find the reciprocal of a number?

To find the reciprocal of a number, simply divide 1 by that number. For example, to find the reciprocal of 4, you would divide 1 by 4, resulting in 1/4.

What are repeating digits?

Repeating digits are numbers that have a repeating pattern of digits after the decimal point. For example, the decimal representation of 1/3 is 0.333333..., where the 3s repeat infinitely.

How do you represent repeating digits in mathematics?

Repeating digits are represented using a bar over the repeating pattern. For example, the repeating decimal 0.333333... would be written as 0.3̅ in mathematical notation.

How do you convert a repeating decimal to a fraction?

To convert a repeating decimal to a fraction, we can use the fact that the decimal represents a fraction with the repeating pattern as the numerator and the number of repeating digits as the denominator. For example, 0.3̅ is equivalent to 3/9, which can be simplified to 1/3.

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