What is the 2012th digit after the decimal point in the decimal expansion of 8/81?
To find the 2012th digit after the decimal point in the decimal expansion of 8/81, we can perform long division.
1) Start by dividing 8 by 81. The quotient will be the part before the decimal point, and the remainder will be used to find the digits after the decimal point.
0.0987654321
81 ) 8.0000000000
0
-------
2) Multiply the remainder (8) by 10 to bring down another digit.
0.0987654321
81 ) 8.0000000000
0
-------
80
3) Divide the new dividend (80) by 81 and continue the process.
0.0987654321
81 ) 8.0000000000
0
-------
80
81
4) Multiply the new remainder (80) by 10.
0.0987654321
81 ) 8.0000000000
0
-------
80
81
-------
800
5) Divide the new dividend (800) by 81.
0.0987654321
81 ) 8.0000000000
0
-------
80
81
-------
800
729
6) Repeat the process until the desired number of digits is reached.
In this case, continuing the division process will eventually lead to a repeating decimal pattern of 0987654321. Since the pattern repeats every 10 digits, we can determine the 2012th digit by finding the remainder when 2012 is divided by 10.
2012 ÷ 10 = 201 remainder 2
The 2012th digit after the decimal point will be the 2nd digit in the repeating pattern, which is 9.
Therefore, the 2012th digit after the decimal point in the decimal expansion of 8/81 is 9.