Mathematical ambiguity when accepting 0.99...=1

You're welcome.Okay, I'm going to try and make things neater this time. First, I'll address the proof that all numbers are equal.Proof That All Numbers Are Equal?The problem comes when taking the root of both sides. Since (-x)^2 = (x)^2, you need to take into account the possibility that one of the sides is negative.Technically, it should go like this:(a - t/2)^2 = (b - t/2)^2±(a - t/2) = ±(b - t/2)Which gives us two equations:a - t/2 = -(b - t/2) & a - t/2 = b - t/2a - t/2 = -b + t/2 & a = ba + b = t One gives you the original equation, and the other gives you something impossible. That's why it's important to consider that some roots may have either positive or negative answers. Not considering that actually is a fallacy.In fact, that very flaw was discussed in your link. (Which you apparently read, so I'm not sure why you included that particular link for an example.)Thus, the thing that makes my algebraic proof better is that it does not have that specific flaw. It does not have multiple possible answers, so that flaw isn't applicable here. If you want to give an example of why the algebraic proof of 0.999...=1 doesn't work, you'd have to give another example of a proof that used the same method to prove something impossible, not a proof with a completely unrelated flaw. If you could disprove something by showing that an unrelated proof didn't work, showing a single flawed proof could disprove absolutely every proof in mathematics.New definition of dsumFor your definition, I'm pretty sure 'countably infinite whole number' isn't possible, as a whole number implies a number under infinity. I'll assume you just meant countably infinite, since that's already related to the set of natural numbers.Also, dsum(9n) is not 9 when n=∞. The reason for this as I said earlier: while you can eventually reach an indefinitely large whole number 9n by adding 9s together, you can never reach infinity. So saying 'no matter how large n is' isn't applicable here because 'no matter how large n is', it cannot equal infinity. Countable infinity should not be treated the same way as whole numbers because it does not have the same properties.I'm also not sure what you mean when you say 'homogeneous digital root'. 'Homogeneous' only applies to functions, differential equations, or numbers. I don't think you're talking about numbers with identical prime factors or functions that scale by c^k when you multiply the input by a constant c because that doesn't work in digital roots for anything other than 9 and k=0, especially once you calculate a root larger than 9 (e.g. dsum(2)=2 dsum(2*11)=4, dsum(2*111)=6, dsum(2*1111)=dsum(10)=1), so I guess you're talking about homogeneity in the chemical or physical sense. I suppose you're considering repeating decimals like 0.333... or 0.999... as homogeneous?And where did that algebraic definition of dr(a) come from? I though the digital root was based on modular arithmetic. (dsum(x)=x%9 unless x%9=0, or x=0, then dsum(x)=9 and dsum(x)=0 respectively.) Do you have a link for the algebraic definition? (By the way, I plugged your equation into a calculator. It didn't work.[1])Furthermore, why are you sure that tan(pi/2) is the same as countable infinity? (It isn't, by the way. tan(pi/2) is actually complex infinity, not normal infinity. I learned that one from wolfram alpha. Besides, I'm not sure normal infinity would be the same as countable infinity, because one describes the size of a set and the other describes a number greater than any whole or real number. And we're talking about setting x to infinity when x can at least be set to any real number, a set which is uncountable.)Even ignoring all that, in your algebra, the way that you got tan(x)=a and tan(y)=b gives you tan(b)=tan(y) doesn't work because tangent function don't work like simple parentheses. It should be easy to see that those are two separate equations that cannot be combined algebraically. Your equation should be tan(∞)=tan(tan(pi/2)) since tan(pi/2)≈∞, not tan(∞)=tan(pi/2). Now, while you say I should take your word with skepticism and write down drF(x), I have no way of doing so. I'm pretty sure the equation you listed isn't correct since the parentheses are mismatched and testing it after fixem them doesn't equal the actual digital root. There's also no link to the correct equation you want me to check. I don't know if just plugging in infinity will work either because both tan(∞) and tan(pi/2) lead to things that aren't whole numbers and will never be without some sort of limit function. The limit function might work here, but I have no idea what it equals without a working equation. I doubt the limit of the algebraic definition of the digital root equals 9, but I can't really tell without that working equation.However, even if your equation worked, I think I've shown that the reasoning behind it is flawed enough that it doesn't apply to this particular situation.Sources:[1] http://www.wolframalpha.com...*0.5%29%282abs%28a%29-1%29%29%2Fpi-5.4%7D%2F9%29+