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AL1L36842y++ for yes
AL1L36842y++ for no
Doesnt matter what we think, because its true lol
I think infinity always does strange things, welcome to math!
Common mistake for math n00bs.
There is a reason that decimal representations are called “approximations.” You always compute using the exact expressions for numbers. If a number is irrational. Like pi, it has no form where it can be written as a quotient of rational numbers. A rational number is defined as a number that can be written as a quotient, in the case of non-imaginary numbers, there exists an expression for the rational number that is a quotient in two integers. If a number is rational, then you can use that property to ascertain the numerator and denominator and perform the computation using the exact form. If a number is irrational, then you are forced to use an approximation and your result will in turn be approximate.
0.999 repeating is in a sense equivalent to 1, as it is an irrational number and does not behave like a rational number. It does not really make sense to talk about repeating decimals as rational numbers, such as 0.333 however they are resulting from rational numbers, so it’s really the users fault that they chose to take the approximation and treat it as exact when it is not.
Hope that clears things up...
maxtnt22yif there is difference between 0.999.. and 1, "0.999.."'s "9" repeating is stop
@runfrodorun I don't know if I agree with the irrational part. This 0.999... = 1 stuff comes from an ambiguity of representation of real numbers, reflects the non-existence of not nil infinitesimals within the real numbers, and is useful when doing things like Cantor diagonalization. 0.333... is still rational, even with the infinite repetition.
@dread-uo in a sense, but only because our intuition is propping up the exact form. Producing a decimal from a fraction is a one way map because we can’t talk about infinite things as finite things
If 0.999 is not irrational then we must be able to write it as a fraction. We cannot do this.
@runfrodorun from an other perspective, lets assume we can store numbers up to infinity, in that case eps(1, 0.999..) would be infinitely small, exactly as small as eps(1, 1), making them virtually the same number
@BindView you’ve made a critical assumption though, and that’s that 0.999 is strictly smaller than 1. I’m asserting that they actually are the same. 0.999 is an approximation of 3 * 1/3 which we know to be one. I know it’s confusing but this is just how decimals behave they are not exact forms for many reasons.
@runfrodorun in fact, you can map a repeating decimal to its corresponding irreducible fraction. All repeating decimals are rational, it's just consistent.
@dread-uo they are all representations of rational numbers, I’m not denying that. I just dislike people saying that infinite decimals are strictly rational, and I do have a masters degree in math. I suppose if you were to define a map from a decimal to an explicit fraction then you could say they are ‘rational’ but I prefer to think of them as representations of rational numbers. Decimal form in higher mathematics does not exist whatsoever. It’s for applied physics and undergraduate Euler calculus imo.
If you don’t mind then, you should produce a rational number that represents 0.9999; really the only appropriate one I can think of is 1, which begs the question are they the same to which I answer yes, because 0.999 is an irrational approximation that essentially represents 1.
Oh, I think I got your point now. It's an interesting way of looking at the problem. The irrational approximation part is a bit strange for me, and I'll give it some more thought :)
Thanks for your patience.
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