0.08 Recurring As A Fraction

0.08 Recurring Decimal:

\[ 0.0\overline{8} = \frac{4}{45} \]

Decimal Input:

Fraction Result:

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1. What Is 0.08 Recurring?

0.08 recurring (written as 0.0\(\overline{8}\)) is a decimal number where the digit 8 repeats infinitely after the first decimal place. It represents the infinite sum 0.08888...

2. Conversion Process

To convert 0.0\(\overline{8}\) to a fraction:

Let x = 0.0\(\overline{8}\)

Then 10x = 0.\(\overline{8}\)

And 100x = 8.\(\overline{8}\)

Subtract: 100x - 10x = 8.\(\overline{8}\) - 0.\(\overline{8}\)

90x = 8

x = 8/90 = 4/45

3. Mathematical Proof

Verification: To verify 4/45 = 0.0\(\overline{8}\):

4 ÷ 45 = 0.08888...

This confirms our conversion is correct.

4. Applications

Uses: Recurring decimals appear in various mathematical contexts including:

Geometric series calculations
Probability theory
Financial mathematics (interest calculations)
Physics (wave equations)

5. Frequently Asked Questions (FAQ)

Q1: Why does 0.08 recurring equal 4/45?
A: The algebraic conversion process demonstrates this equivalence mathematically.

Q2: Can all recurring decimals be converted to fractions?
A: Yes, any repeating decimal can be expressed as a fraction using similar algebraic methods.

Q3: How is this different from terminating decimals?
A: Terminating decimals have finite digits while recurring decimals have infinite repeating patterns.

Q4: What's the simplest form of 8/90?
A: Both numerator and denominator can be divided by 2, giving 4/45.

Q5: Are there other fractions that produce similar patterns?
A: Yes, for example 1/9 = 0.\(\overline{1}\), 2/9 = 0.\(\overline{2}\), etc.