When we look at numbers with many decimal places, particularly in recurring decimals, we can explore patterns in the sequence. In this article, we will find the 300th digit of the decimal representation of 0.0588235294117647. This number is an interesting one, as it is a repeating decimal.
Understanding the Number
The given decimal 0.0588235294117647… is the result of dividing 1 by 17:
1÷17=0.0588235294117647‾1 ÷ 17 = 0.\overline{0588235294117647}The notation 0588235294117647‾\overline{0588235294117647} means that the sequence 0588235294117647 repeats endlessly after the decimal point.
Identifying the Pattern
- The repeating block in the decimal is 0588235294117647.
- This block has 16 digits.
0.05882352941176470588235294117647…0.05882352941176470588235294117647\ldotsSince the decimal repeats every 16 digits, any digit at a position beyond the 16th can be determined by identifying where it falls within the repeating pattern.
Finding the 300th Digit
To find the 300th digit, we need to determine which digit it corresponds to in the repeating block of 16 digits.
We do this by dividing 300 by 16:
300÷16=18 remainder 12300 ÷ 16 = 18 \, \text{remainder} \, 12The remainder tells us which digit in the repeating block corresponds to the 300th position. Since the remainder is 12, the 300th digit will be the 12th digit of the repeating sequence 0588235294117647.
Determining the 12th Digit
The repeating sequence is:
05882352941176470588235294117647The 12th digit in this sequence is 4.
Conclusion
Thus, the 300th digit of the decimal representation of 0.0588235294117647 is:
4\boxed{4}
Final Thoughts
This exercise demonstrates how to find specific digits in repeating decimals by using simple division and pattern recognition. The key is to identify the repeating block and determine where the desired position falls within it.
