Finding the Digit at the 100th Place in the Decimal Representation of 6/7: A Comprehensive Guide
When dealing with decimals, especially those that have repeating patterns, it can be quite intriguing to determine specific digits within these sequences. This article will guide you through the process of finding the 100th digit in the decimal representation of 6/7.
Understanding the Decimal Representation of 6/7
The decimal representation of 6/7 is 0.857142857142... and this sequence of '857142' repeats indefinitely. Recognizing this pattern is crucial to solving the problem efficiently.
Step-by-Step Solution
1. **Identify the Repeating Sequence**:
The repeating sequence in the decimal representation of 6/7 is '857142', which is 6 digits long.
2. **Determine the Position in the Repeating Cycle**:
To find the 100th digit, we need to determine its position within the repeating sequence. We do this by dividing 100 by 6.
100div616, text{remainder}, 4.
3. **Identify the Digit in the Sequence**:
The remainder indicates that the 100th digit corresponds to the 4th digit in the repeating sequence '857142'.
1st digit: 8 2nd digit: 5 3rd digit: 7 4th digit: 1 5th digit: 4 6th digit: 2
Therefore, the 4th digit, which is the 100th digit in the decimal representation of 6/7, is 1.
Alternative Methods and Verification
1. **Using Division**:
Perform the division 6 div 7 to see the pattern arising from the decimal representation. This confirms that the repeating sequence is '857142', and by dividing 100 by 6, we find that the 100th digit is the 4th digit in this sequence, which is 1.
2. **Mathematical Insight**:
Consider the number 100/6. The result is 16 with a remainder of 4, confirming that the 100th digit is the 4th digit in the repeating sequence.
3. **Using Wolfram Alpha**:
Wolfram Alpha provides a more formal way of confirming this by performing a specific calculation. The expression leftlfloor frac{6}{7} times 10^{100} rightrfloor bmod 10 equiv 1 confirms that the 100th digit is indeed 1.
Conclusion
In conclusion, the process of finding the digit at the 100th place in the decimal representation of 6/7 involves recognizing the repeating sequence and using modular arithmetic to pinpoint the exact digit within that sequence. This guide should help you solve similar problems efficiently and accurately.