1. Understanding 0.0099... as a Fraction

The repeating decimal 0.009900990099... is exactly equal to the fraction 1/101. This is a special case of repeating decimals where the repeating block "0099" corresponds to this simple fraction.

2. Mathematical Proof

Here's how we can prove that 0.0099... = 1/101:

This shows the exact fractional equivalent of the repeating decimal.

3. Practical Applications

Details: This conversion is useful in various mathematical calculations, especially when exact fractional values are needed rather than decimal approximations.

4. Interesting Properties

Pattern Recognition: The decimal expansion of 1/101 shows an interesting repeating pattern of 0099 that makes it memorable and easy to identify.

5. Frequently Asked Questions (FAQ)

Q1: Why does 0.0099... equal 1/101?
A: This is a mathematical identity that can be proven using algebra as shown above. The repeating pattern corresponds exactly to this fraction.

Q2: Can all repeating decimals be converted to fractions?
A: Yes, any repeating decimal can be converted to an exact fraction using similar algebraic methods.

Q3: How can I verify this conversion?
A: You can verify by dividing 1 by 101 using long division, which will produce the repeating decimal 0.009900990099...

Q4: Are there other fractions with similar patterns?
A: Yes, fractions with denominators like 1001, 10001, etc., produce similar repeating patterns in their decimal expansions.

Q5: Why is this fraction important?
A: While not commonly used in everyday calculations, understanding such conversions helps in advanced mathematics and number theory.