Yes, \( 0.999\ldots \) (repeating) is equal to 1. This can be shown using a simple algebraic proof:

1. Let \( x = 0.999\ldots \).
2. Multiply both sides by 10: \( 10x = 9.999\ldots \).
3. Subtract the original equation from this result: \( 10x - x = 9.999\ldots - 0.999\ldots \).
4. Simplifying, you get \( 9x = 9 \).
5. Dividing both sides by 9: \( x = 1 \).

Thus, \( 0.999\ldots = 1 \).

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### Related Questions:
1. What is a repeating decimal, and how is it different from a terminating decimal?
2. How does the concept of limits in calculus relate to \( 0.999\ldots = 1 \)?
3. Can other repeating decimals be simplified to a fraction?
4. How does this concept work in different number systems, like binary?
5. What are some real-world applications where repeating decimals are important?

**Tip:** Understanding why \( 0.999\ldots = 1 \) helps strengthen your grasp of concepts like limits and the real number system.

Explore why 0.999 repeating is equal to 1 through a detailed algebraic proof. This problem delves into decimal representation and the concept of repeating decimals, essential for advanced high school mathematics.

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