What is 0.8 repeating as a fraction
The Correct Answer and Explanation is:
ChatGPT said:
Correct Answer:
The decimal 0.8 repeating (written as 0.8‾0.\overline{8}0.8) is equal to the fraction 89\frac{8}{9}98.
Explanation
To understand why 0.8‾=890.\overline{8} = \frac{8}{9}0.8=98, we need to explore how repeating decimals convert into fractions.
Step-by-Step Conversion:
- Let x=0.8‾x = 0.\overline{8}x=0.8, which means the digit 8 repeats forever:
x=0.88888…x = 0.88888\ldotsx=0.88888… - Multiply both sides by 10 to move the decimal point one place to the right:
10x=8.88888…10x = 8.88888\ldots10x=8.88888… - Now subtract the original equation from this new equation: 10x=8.88888…−x=0.88888…9x=8\begin{align*} 10x &= 8.88888\ldots \\ – x &= 0.88888\ldots \\ \hline 9x &= 8 \end{align*}10x−x9x=8.88888…=0.88888…=8
- Solve for xxx:
x=89x = \frac{8}{9}x=98
This shows that the repeating decimal 0.8‾0.\overline{8}0.8 is exactly equal to 89\frac{8}{9}98.
Why This Works:
Repeating decimals have patterns that make them expressible as fractions. When a decimal repeats, multiplying by a power of 10 helps align the repeating digits. Subtracting removes the repeating part, leaving a solvable equation.
The method works for any single-digit repeating decimal:
- 0.1‾=190.\overline{1} = \frac{1}{9}0.1=91
- 0.2‾=290.\overline{2} = \frac{2}{9}0.2=92
- …
- 0.8‾=890.\overline{8} = \frac{8}{9}0.8=98
- 0.9‾=99=10.\overline{9} = \frac{9}{9} = 10.9=99=1
This pattern reveals that repeating decimals represent rational numbers—numbers that can be written as ratios of integers.
Final Note:
Understanding how repeating decimals become fractions helps clarify the nature of rational numbers and shows that what looks like an endless decimal is actually a precise, finite value in fractional form. So, 0.8‾=890.\overline{8} = \frac{8}{9}0.8=98 is exact and not an approximation.
