Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.
To express 811 as a repeating decimal, we divide 811 by 999. The result is 0.8118118118... and so on. We can mark the cycle of repeating digits by placing a bar over the digits, giving us the decimal representation: 0.811̅.
To convert the number 811 into a repeating decimal, we can use a process called long division. Let's begin:
1) Divide 811 by the divisor, which is 9.
9|811
-72
-----
91
2) Bring down the next digit (which is 1), and bring down a decimal point.
9|811
-72
-----
91
9
3) Divide 91 by 9.
9|811
-72
-----
91
9
-81
-----
101
4) Bring down the next digit (which is 1).
9|811
-72
-----
91
9
-81
-----
101
9
5) Divide 101 by 9.
9|811
-72
-----
91
9
-81
-----
101
9
-9
-----
2
6) Bring down the next digit (which is 2).
9|811
-72
-----
91
9
-81
-----
101
9
-9
-----
2
-0
-----
20
7) Divide 20 by 9.
9|811
-72
-----
91
9
-81
-----
101
9
-9
-----
2
-0
-----
20
-18
-----
20
8) Bring down the next digit (which is 0).
9|811
-72
-----
91
9
-81
-----
101
9
-9
-----
2
-0
-----
20
-18
-----
20
-18
-----
2
At this point, we can see that the division does not repeat any further. So the repeating decimal representation of 811 is 90.111..., with the digit sequence 111 repeating. In other words, we can write it as:
811 = 90.111... (the bar is placed on top of the repeating digits 111)