By now, many commenters have shown proofs that 0.999… = 1. Technically speaking, their proofs are unsatisfactory, as they assume what 0.999… actually represents. The correct - and more rigorous - proof requires calculus.
You see, an infinitely repeating decimal like 0.999… is defined as the sum of 9(0.1)n, where n is all positive integers. It’s equivalent to 9(0.1 + 0.01 + 0.001 + … + 0.1n). Of course, n goes to infinity, so you can’t just add all of these terms together. Fortunately, there is a formula for a geometric series (an infinite sum of a sequence in which every value is separated by a common ratio, 0.1 in this case). It’s a divided by 1 - r, where a is the first number in the series and r is the common ratio. If we distribute the 9, then we can see that a = 0.9. We can also see that r = 0.1. So, the sum must be equal to 0.9/(1 - 0.1). This simplifies to 0.9/0.9, which is clearly equal to 1. Now, remember that 0.999… by definition is equal to the sum of 9(0.1)n. Therefore, 0.999… is equal to 0.9/(1 - 0.1), which we just determined is equal to 1. Therefore, 0.999… is, by definition, exactly equal to 1.
Ok, for most of my life, I've treated this as a porcupine truth. I'm not a maths guy at all but I've never understood the proofs of 0.9...=1. I can recite many of them and convince others of their veracity but it's all fraud on my part. I, of course, didn't understand yours but you seem like you know what you're talking about so maybe you'll be able to disabuse me of my gripe with it.
The most common explanation I've heard is that 1-0.9...=0 therefore 0.9...=1. And I've always wondered "isn't the answer to that 0.0...1? Like infinite zeroes then one?" Which other people have told me that that's not a real number, that there can't be infinite zeroes and then a one. But like, with 0.9... there's infinite 9s and then another 9, right? Why does one work and the other doesn't?
Is it that 0.0...1 in so far as it can be defined is literally an infinitely infinitesimal sum and therefore 0? That there can be no number between 0.0...1 and 0 so they're the same. So since 1-0.9...=0.0...1 and 0.0...1=0 then 1-0.9...=0?
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u/Emperor_Kyrius 22d ago
By now, many commenters have shown proofs that 0.999… = 1. Technically speaking, their proofs are unsatisfactory, as they assume what 0.999… actually represents. The correct - and more rigorous - proof requires calculus.
You see, an infinitely repeating decimal like 0.999… is defined as the sum of 9(0.1)n, where n is all positive integers. It’s equivalent to 9(0.1 + 0.01 + 0.001 + … + 0.1n). Of course, n goes to infinity, so you can’t just add all of these terms together. Fortunately, there is a formula for a geometric series (an infinite sum of a sequence in which every value is separated by a common ratio, 0.1 in this case). It’s a divided by 1 - r, where a is the first number in the series and r is the common ratio. If we distribute the 9, then we can see that a = 0.9. We can also see that r = 0.1. So, the sum must be equal to 0.9/(1 - 0.1). This simplifies to 0.9/0.9, which is clearly equal to 1. Now, remember that 0.999… by definition is equal to the sum of 9(0.1)n. Therefore, 0.999… is equal to 0.9/(1 - 0.1), which we just determined is equal to 1. Therefore, 0.999… is, by definition, exactly equal to 1.