This was not done to reopen the discussion, but simply to supply what I thought was interesting and what was requested. I will be happy to discuss the mathematics or methodology involved, but it is not any further attempt to "prove someone wrong".
All you are showing is that .9.. is less than 1 because you have a remainder. Do the same thing, and you can show .8.. = 1, if you ignore the rules of division. When you have an equal quantity that yields no remainder, you use it first. 1 divided by 1 is 1
Here is another little thing I came up with the last time we went round and round on this. Let us assume .9… < 1. Then there must be some Z Є ℝ such that .9… < Z < 1 by the theorem that between any two distinct Reals there is a rational number. By theorem that any repeating decimal number is a rational number, and the theorem that the sum of a rational and an irrational is irrational, Z = .9… + q where q Є ℝ and q is rational. ** The smallest non-zero value for q will be of the form q = 1/10^k where k Є ℕ . Anytime ∑ is written, it will be for (n from 1-> ∞ ) .9… = .9 + .09 + .009 + .0009 + … = 9/10 + 9/100 + 9/1000 + … = 9/10^1 + 9/10^2 + 9/10^3 + … = ∑ 9/10^n Z = ∑ 9/10^n + q = ∑ 9/10^n + 1/10^k Let k = 1 Z = ∑ 9/10^n + 1/10 = 1/10 + 9/10 + 9/100 + 9/1000 + … = 10/10 + 9/100 + 9/1000 + … = 1 + 9/100 + 9/1000 + … = 1 + ∑ 9/10^(n+1) This would make Z > 1 which violates our original assumption, there for, q cannot be 1/10. Now let k any number in ℕ and k > 1. Z = ∑ 9/10^n + 1/10^k = 1/10^k + 9/10 + 9/100 + 9/1000 + … + 9/10^(k-1) + 9/10^k + 9/10^(k+1) + … = 9/10 + 9/100 + 9/1000 + … + 9/10^(k-1) + 10/10^k + 9/10^(k+1) + … = 9/10 + 9/100 + 9/1000 + … + 9/10^(k-1) + 1/10^(k-1) + 9/10^(k+1) + … = 9/10 + 9/100 + 9/1000 + … + 9/10^(k-2) + 1/10^(k-2) + 9/10^(k+1) + … [Continue until a total of k iterations are completed] = 9/10 + 1/10 + 9/10^(k+1) + … = 1 + 9/10^(k+1) + … = 1 + ∑ 9/10^(n+k+1) Thus Z > 1 This implies that for any q > 0, Z > 1, which goes against our original assumption that .9… < 1. Therefore, .999… cannot be less than 1.
0.8.. = 1 True or false? Because I can use your example as “proof” if you claim yours is as well. That’s why.
Using Alan methodology that says I don't have to use an equal quantity that yields no remainder first 1|1 = 0.8 (ignore the remainder) 0.88 (ignore the remainder) 0.888 (ignore the remainder) .. 0.8.. (ignore the remainder) Or, exactly what you did above.
All this also requires that any repeating decimal number be rational. 0.0.. is repeating decimal number. And it is not rational, because there is no non zero denominator q p/q that produces it.
Please look again, I am not ignoring the remainder. The remainder is clearly seen on the left hand calculation.
But, logically, and in mathematics: 0.01 [exists] 0.0000001 [exists] 0.000000000000000000000000001 [exists] 0.0000000000000000000000000000000000000000000001 [exists] 0.[a billion zeros]1 [exists] 0.[a billion billion billion zeros]1 [exists] 0.[a billion billion trillion zeros]1 [exists] 0.[a trillion trillion trillion billion trillion trillion billion trillion zeros]1 [exists] And then math says 0.[n]1 Whoa whoa whoa. Doesn't exist, it's zero. Illogical. But, that's Math.
What is n? Don't say infinity because that isn't a number and nothing can be in the "infinitieth place" after the decimal.
Valid, but it still requires that all repeating digits be labeled as rational. Which is a rule of mathematics. It's still circular.
Why do you think that? A monkey, writing zeros, and then a 1 after, writing for infinite time... how many zeros has he written?
Mathematics starts with a series of axioms that we must assume to be true because they cannot be proven. Everything else derives from those axioms. Don't like it? Awesome, devise your own system using your own axioms. But any such system will be 'circular'. But we use it because it works. It makes sense of the natural world.
You don't handle the remainder, and you have a 8 after a string a 9s, which you say is also invalid. Your method would look more like : 0.9..8 than anything else.
It does, generally. But not here. It is illogical here, and demonstrably so. I've already provided them. They work.
The number on the right hand side where I am adding the values is a static number. .998 After infinite iterations of this, you would end up with .999...
They result in 0 < 0 as I showed a few pages ago. Your method results in contradictions. I am willing to continue working on your method, as I find the exercise of it fun (I am weird that way).
