Arrange the following real numbers in order from greatest to least:
0.8, 0.8(repeating 8), 0.89(repeating 89), 0.889( repeating the second 8 and 9), and 0.88.
I think I got it, but just want to make sure.
Not quite the order is
0.8000000
0.8800000
0.8888888
0.8898989
0.8989898
To arrange the real numbers in order from greatest to least, we can start by comparing the whole numbers before the decimal point.
The given numbers are:
0.8, 0.8(repeating 8), 0.89(repeating 89), 0.889(repeating the second 8 and 9), and 0.88.
First, let's compare the whole numbers:
0.8 < 0.88 (since 8 is smaller than 8)
0.8 < 0.889 (since 8 is smaller than 8)
0.8 < 0.89 (since 8 is smaller than 8)
0.8 < 0.8(repeating 8) (since 8 is smaller than 8)
Next, let's compare the repeating decimal parts:
0.8(repeating 8) < 0.88 (since 8 is smaller than 8)
0.8(repeating 8) < 0.889 (since 8 is smaller than 8)
0.8(repeating 8) < 0.89 (since 8 is smaller than 8)
Finally, let's compare the non-repeating decimal parts:
0.88 < 0.889 (since 8 is smaller than 9)
Putting everything together, the numbers arranged from greatest to least are:
0.889(repeating the second 8 and 9) > 0.89(repeating 89) > 0.8(repeating 8) > 0.88 > 0.8
Therefore, the correct order is:
0.889(repeating the second 8 and 9) > 0.89(repeating 89) > 0.8(repeating 8) > 0.88 > 0.8
To arrange the given real numbers from greatest to least, you can compare the decimal parts of the numbers. Let's consider each number:
1. 0.8: This number is straightforward, and its decimal part is just 0.8.
2. 0.8(repeating 8): This number has an infinite string of 8s. Since we only need to compare the first decimal places, we can consider this as 0.8888... You can think of this number as x and solve the equation 10x = 8.888... by subtracting x from 10x. This gives 9x = 8.888... - 0.888... which simplifies to 9x = 8. (Only the decimal parts were subtracted.) Solving for x, we find that x = 8/9 = 0.888... So, the decimal part of this number is 0.888...
3. 0.89(repeating 89): Similar to the previous number, let x be this number. Considering it as x, we need to solve the equation 100x = 89.8989... by subtracting x twice and simplifying. This gives 99x = 89.8989... - 0.8989... which simplifies to 99x = 89. So, x = 89/99 = 0.8989... So, the decimal part of this number is 0.8989...
4. 0.889(repeating the second 8 and 9): As before, let x be this number. Considering it as x, we need to solve the equation 1000x = 889.98989... by subtracting x thrice and simplifying. This gives 999x = 889.98989... - 0.98989... which simplifies to 999x = 889. So, x = 889/999 = 0.889889... So, the decimal part of this number is 0.889...
5. 0.88: This number is straightforward and has a decimal part of just 0.88.
Now, let's arrange the numbers based on their decimal parts:
0.8989... > 0.889... > 0.888... > 0.88 > 0.8
Therefore, the correct order from greatest to least is:
0.89(repeating 89) > 0.889(repeating the second 8 and 9) > 0.8(repeating 8) > 0.88 > 0.8