To determine how many digits long the repeating cycle of a repeating decimal would be if the denominator of the rational number is 7, we can use the concept of modular arithmetic.
When dividing any positive integer by 7, there are only 7 possible remainders: 0, 1, 2, 3, 4, 5, and 6. Therefore, in any repeating decimal fraction with a denominator of 7, the possible remainders in the repeating cycle are also limited to these 7 values.
Let's calculate the decimal representation of some fractions with a denominator of 7:
- 1/7 = 0.142857142857142857...
- 2/7 = 0.285714285714285714...
- 3/7 = 0.428571428571428571...
- 4/7 = 0.571428571428571428...
- 5/7 = 0.714285714285714285...
- 6/7 = 0.857142857142857142...
From the above decimal representations, we can see that the repeating cycle is 6 digits long for all fractions with a denominator of 7.
Therefore, the repeating cycle of a repeating decimal with a denominator of 7 would be 6 digits long.
How many digits long would the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
2 answers
Convert 0.27¯¯¯¯¯
to a rational number in simplest form
to a rational number in simplest form
1 answer