The proof that 0.999... is equal to 1 is based on the concept of limits in calculus. The decimal representation of 0.999... is an infinite series that converges to the number 1. This means that as we add more and more 9s to the end of the decimal, the value gets closer and closer to 1. In the limit, the value of 0.999... is equal to 1.
The difference in decimal representations is due to the fact that we are using a finite number of digits to represent an infinite number. In reality, both 0.999... and 1 are just different ways of representing the same number, which is 1.
No, 0.999... is not an approximation of 1. It is exactly equal to 1, as shown by the mathematical proof using limits.
The concept of limits and infinite series can also be applied to other repeating decimals. For example, the repeating decimal 0.333... is equal to the fraction 1/3, and the repeating decimal 0.142857... is equal to the fraction 1/7. This concept helps us understand that repeating decimals are just different ways of representing rational numbers.
The never-ending 0.999... = 1 debate is important because it challenges our understanding of numbers and the concept of infinity. It also highlights the power and usefulness of mathematical proofs in providing concrete answers to seemingly abstract questions. Understanding this concept can also help in other areas of mathematics, such as calculus and real analysis.