![]() ![]() However, since one of the digits was shifted left across the decimal, that means that, on the right of the decimal, there is only ω – 1 digits! This is still an infinite number, but it is a different infinite number. If we multiply the number by 10, then the value becomes 9.999…. Let’s be more specific about how many digits that 0.999… has. We will use the Greek symbol ω to represent whatever specific infinity of digits that this number has after the decimal point. ![]() Once we understand this, the problem with the original proof becomes clear. The obvious problem with this proof is that it is considering a whole class of numbers to be equal to each other, rather than merely a member of a class. This is what we tend to do with infinite values-even though they are a class of numbers, we treat them all as if they were the same value. For instance, consider the following obviously false proof: Infinity is not a single number, but rather a class of numbers-like odd or even. Just like even and odd numbers, we can make some rules about their usage: odd + odd = even even + even = even even + odd = odd. However, we can’t simply substitute odd and even for the values they represent. ![]() has the same number of digits as 9.999… It is true that they both have an infinite number of digits, but they are not necessarily the same number of infinite digits. The problem here is the assumption that, after multiplying by 10, 0.999…. This problem is exacerbated by much mathematical notation. People often will use ellipses to indicate that something goes on “for infinity”, whether a series notation or just a continuation of a decimal expansion. Therefore, they feel that the original set, and a set with just one more or one less added, are the same set. Therefore, you get “proofs” like the following “proof” for why 0.9999… = 1: They believe that 2 * infinity = infinity. However, using that logic can quickly lead to contradictions. Therefore, people have the concept that if I have two infinities, then I still have the same number. Because infinity is bigger than all possible natural numbers, people assume that it is bigger than any number, and therefore there is nothing beyond infinity. The first problem people have with infinity is that they treat it as if it were a single value. The concept of infinity has plagued a great many proofs, both formal and informal. I think that there are two foundational problems at play in most people’s thinking about infinity that causes issues. Share Facebook Twitter LinkedIn Flipboard Print arroba Email ![]()
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