Asked by .....

Convert 0.8 to a rational number
0.8 is a repeating decimal
Answered by .....
0.8 repeats
Answered by .....
yes i did mean 0.888...
Answered by GPT-4o mini
The decimal 0.8 can be expressed as a rational number. It is not a repeating decimal; it is a terminating decimal.

To convert 0.8 to a rational number:

1. Recognize that 0.8 can be written as \(\frac{8}{10}\).
2. Simplify \(\frac{8}{10}\) by dividing the numerator and denominator by their greatest common divisor, which is 2.

\[
\frac{8 \div 2}{10 \div 2} = \frac{4}{5}
\]

Therefore, the rational form of 0.8 is \(\frac{4}{5}\).
Answered by GPT-4o mini
If the decimal is \(0.888...\), which is a repeating decimal, we can convert it to a rational number as follows:

Let \(x = 0.888...\).

1. Multiply both sides by 10 to shift the decimal point:
\[
10x = 8.888...
\]

2. Now, you have two equations:
\[
x = 0.888...
\]
\[
10x = 8.888...
\]

3. Subtract the first equation from the second:
\[
10x - x = 8.888... - 0.888...
\]
\[
9x = 8
\]

4. Solve for \(x\):
\[
x = \frac{8}{9}
\]

Thus, if \(0.8\) is meant as a repeating decimal (\(0.888...\)), its rational form is \(\frac{8}{9}\). If you meant \(0.8\) as a non-repeating number, the rational form is \(\frac{4}{5}\). Please clarify if you meant \(0.888...\).
Answered by GPT-4o mini
Great! Since you meant \(0.888...\) (which represents the repeating decimal), we already found that:

\[
x = 0.888...
\]

1. Multiply both sides by 10:
\[
10x = 8.888...
\]

2. Subtract the first equation from the second:
\[
10x - x = 8.888... - 0.888...
\]
\[
9x = 8
\]

3. Solve for \(x\):
\[
x = \frac{8}{9}
\]

Therefore, the rational representation of \(0.888...\) is \(\frac{8}{9}\).