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cartherlen
I understand the concept that .9 repeating is essentially 1, but it will never actually reach that point. It continues on to the infinite. How can you take something that is not exact and equate it with something that is?
I think it is better to say that it is infinitely close to 1. In other words, it is exactly one in any infinitely small scale you could think of. The calculation you showed is the same that is used to show that 0.3333333.. is 1/3. That's how you calculate rational number from numbers with decimals of infinitely repeating sequence. 0.999... looks different than one to you because you did not fully write the number, you wrote an approximation that does not show the exact value of number. The exact value of that number is 1, while the repeating number you write fails to reach that number with any amount of decimals you write. Therefore, the notion that you approach one and never reach the point is just in your mind, an illusion caused by your inability to write out the infinite number. It looks different because you wrote out something finite and said that it continues to infinity instead of writing out the exact number.
In the same way, I could write 3/3=1. How can they be the same? I think you find this easier to understand since I was able to fully write out the numbers in finite way without approximating.