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Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?
by hilbertseries 6y ago
Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?
- dacohenii 6y agoI'm pretty sure it's a real number in that it falls within ℝ (the set of all real numbers).
- RHSeeger 6y agoI doesn't, because 0.666....7 is not a valid representation for a number. The ... means goes on forever, and you cannot put something "after forever". It's not different than saying "0.0j" is a real number in base 10; it's not, that string does not represent a number in our number base 10 number system.
- hilbertseries 6y agoThere’s not a way to define it, typically you’d define .666... as the sum of 6/10^n from 1 to infinity. This decimal representation does not terminate, so you can’t put a 7 at the “end of it” because there is no end.
- OscarCunningham 6y agoIt's unfair to ask 'Can you define it?' to someone not educated in real analysis because they don't know which definitions you'll accept and which you won't. They already think that '0.666...7' is a valid definition.
- klodolph 6y agoYou would define it as a digit sequence, using ω + 1 as the indexing set. I would consider it to be most naturally an element of the hyperreal numbers, although it is also contained in smaller extensions of the real numbers.
- hilbertseries 6y agoSure, you could define some sort of number system that has an infinite point digit. I doubt that’s what the original poster had in mind though.
- klodolph 6y agoThese number systems already exist, I’m not just making them up. If someone says 0.666…7 then you have a couple different ways you can take the discussion. You can say, “No! Real numbers don’t work like that!” or you can talk about what number systems would look like if you can do that. It turns out that there’s a lot to learn from the alternative number systems, including formulations of calculus without limits that are easier to understand from an intuitive perspective, yet equally rigorous. The field is called “nonstandard analysis”. It’s not taught in college.
- TheRealPomax 6y agoCan you explain what you mean with "real" in that sentence? Because in the context of maths, a real number is "a number in ℝ", which this absolutely qualifies for. Whereas in plain English the term doesn't really have a clear definition. You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long, but trivially "made" (such as π, which just requires grabbing a compass and drawing a circle) The bit you should be wondering about is why 0.666...6 and 0.666...7 are the same number: infinities cause digits written on paper (or a computer screen) to look like a kind of number that they're not. The two fractions (numbers in ℚ) 0.6666 and 0.66667 are 0.00001 apart, but the two reals (numbers in ℝ) 0.666...6 and 0.666...7 are 0.000...1 apart. That looks like a tiny tiny fraction, but it's not a fraction, it's an infinite number of zeroes, and thanks to that, this number, while it looks like a fraction, is just a silly way to write zero. So thanks to infinities, the most-definitely-not-a-fraction number that we write as 0.666... is the same as the most-definitely-not-a-fraction number 0.666...(some numbers here). The difference between the two is zero. Infinities are fun. And difficult. But also fun.
- hilbertseries 6y agoInfinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is meaningless with respect to the real numbers.
- brlewis 6y agoI don't think it's a real number, but if it was this would be yet another intuitive proof that 0.999... = 1.