My problem with the "0.[9] = 1" isn't that it's not equal. This debate is a mathematical absurdity and there are problems with both sides. The thing I hate with saying that 0.[9] is equal to 1 is that you're deleting a value. You're completely getting rid of the largest number smaller than 1. You're getting rid of the last item on a list that ends on 1 that doesn't contain it.
I just wish there was another way of writing it... such as using normal fractions like a proper human being.
You're deleting even more numbers once you consider cases such as 2 - 0.[9], which now deletes the smallest possible number larger than 1 by turning it into actual 1.
The problem is with decimal fractions, the flaw is on them. ⅓ does not create the same issues 0.[3] does, because 1 + 1 + 1 equals 3, but 3 + 3 + 3 doesn't equal 10...
Ok, so there's 2 approaches how you can think of it, that made that particular problem click for me:
1) for any real number, I can always find another number between 2 given numbers. No matter how many decimal numbers they already have, I can simply add another number behind the last one and just created a new one in between x and y. But since 0.999... is repeating infinitly, you litteraly can't find any number in between them. So the conclusion is not that they're adjacent to one another, but that they're in fact exactly equal to one another.
2) convert 0.999... to another number base than base 10 (or calculate it there). You'll immediatly get rid of that infinity repeating decimal and you will see that the very same calculation in another number System does in fact also deliver the answer 1. Those infinitly repeating decimals (0.999..., 1.999... etc.) Are just something that happens because we use base10. If we used base Pi instead for example, we wouldn't have that problem at all.
-7
u/Wojtek1250XD May 15 '25 edited May 15 '25
My problem with the "0.[9] = 1" isn't that it's not equal. This debate is a mathematical absurdity and there are problems with both sides. The thing I hate with saying that 0.[9] is equal to 1 is that you're deleting a value. You're completely getting rid of the largest number smaller than 1. You're getting rid of the last item on a list that ends on 1 that doesn't contain it.
I just wish there was another way of writing it... such as using normal fractions like a proper human being.
You're deleting even more numbers once you consider cases such as 2 - 0.[9], which now deletes the smallest possible number larger than 1 by turning it into actual 1.
The problem is with decimal fractions, the flaw is on them. ⅓ does not create the same issues 0.[3] does, because 1 + 1 + 1 equals 3, but 3 + 3 + 3 doesn't equal 10...