Understanding the Decimal Representation of 1/11: Finding the 66th Digit
Exploring the decimal representation of fractions can be both fascinating and enlightening. Today, we want to delve into the specific case of the fraction 1/11. We will walk you through the process of identifying the 66th decimal digit of its repeating sequence. This detailed exploration will also cover the principles of modulo arithmetic and its application in solving such problems.
The Decimal Representation of 1/11
Let's begin by determining the decimal representation of 1/11. When we perform the division of 1 by 11, we obtain a repeating decimal:
1/11 0.090909...
We can denote this repeating sequence as:
0.09
Identifying the Position of the 66th Decimal Digit
Given the repeating sequence 09, we can use mathematical methods to determine the 66th decimal place. Let's break it down step by step:
Understanding the Repeating Block: The repeating block for 1/11 is 09, which has a length of 2. This means the sequence repeats every 2 digits.
Modulo Arithmetic: To find the position of the 66th digit within the repeating block, we calculate:
66 mod 2 0
When the result of the modulo operation is 0, it indicates that the 66th digit corresponds to the last digit of the repeating block.
Determining the Digit: Since the last digit of the repeating block 09 is 9, the 66th decimal digit must be 9.
Pattern Recognition in Repeating Decimals
It’s also insightful to observe the pattern in the decimal representation of 1/11. As you can notice, every odd-placed digit is 0, and every even-placed digit is 9. This pattern can be summarized in the following manner:
Odd-placed digits (1st, 3rd, 5th, etc.): 0
Even-placed digits (2nd, 4th, 6th, etc.): 9
Based on this pattern, we can deduce that the 66th decimal digit is 9, as 66 is an even number.
Verification Through Calculator and Long Division
To further validate our findings, we can use both a calculator and long division:
Calculator Method: Using a scientific or graphing calculator, you can compute the decimal expansion of 1/11, which will show 0.090909...
Long Division Method: Performing long division of 1 by 11 manually will yield the same repeating decimal sequence: 0.090909...
In both cases, the repeating block of digits is 09, confirming our earlier findings.
Conclusion
In conclusion, the 66th decimal digit in the decimal representation of 1/11 is 9. This result is obtained by recognizing the repeating sequence and applying modulo arithmetic to determine the position within the block. Whether you use a calculator, long division, or simple mathematical reasoning, the answer remains consistent and clear.
Technical Keywords
Decimal representation: A way of expressing numbers using a base-10 system, where the position of a digit determines its value.
Repeating decimal: A decimal number that continually repeats the same sequence of digits after the decimal point.
Modulo arithmetic: A type of arithmetic in which numbers "wrap around" upon reaching a certain value, the modulus.