The Altera Centauri collection has been brought up to date by Darsnan. It comprises every decent scenario he's been able to find anywhere on the web, going back over 20 years.
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Call To Power 2 Cradle 3+ mod in progress: https://apolyton.net/forum/other-games/call-to-power-2/ctp2-creation/9437883-making-cradle-3-fully-compatible-with-the-apolyton-edition
Please explain what injective means so that your post will make sense to a normal person. My web browser thinks it's not spelled correctly and has put a red line under it.
An injective function has a unique output for each input. So f(x)=sqrt(x) is injective, because no two x-values will produce the same y value. f(x)=x^2, however, is not injective, because x=-2 and x=2 produce the same y-value. What Hauldren is saying is that the base 10 number system is more like x^2 than sqrt(x).
Note: I could be completely wrong, because I know **** all about math.
.999... is a rational number? Please express it as a ratio of two integers.
1:1, but that's what we're trying to prove. A number that you can write as a repeating decimal fraction is a rational number. This is the definition I'm using.
As HC says, you can write 1 either as 1.(0) or 0.(9).
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Gribby's acting like an idiot. The proof is constructive. If you give me a c>0, I can give you a terminating sequence 0.99999... such that 1 - this sequence is less than c. Thus, the difference between 0.999... and 1 is less than any number greater than 0 (and it's >=0) therefore it is 0.
This is a standard proof technique in any real analysis course.
Gribby's acting like an idiot. The proof is constructive. If you give me a c>0, I can give you a terminating sequence 0.99999... such that 1 - this sequence is less than c. Thus, the difference between 0.999... and 1 is less than any number greater than 0 (and it's >=0) therefore it is 0.
This is a standard proof technique in any real analysis course.
Thank you for providing a proof that onodera didn't provide
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But the English language phrase 'Does .999 repeating equal 1?' is not necessarily the same thing as the mathematical phrase '.999... = 1'. These are different and distinguishable sets and the fact that the English language phrase can have the same value as the mathematical phrase does not mean that it cannot have other values where the phrase 'equal' has a looser meaning.
Specifically, with regard to cryptography, '.999...' would not hash to the same number as '1' would. Even more simply if I translate '.999...' into Japanese, it would not come out as the kana for '1', despite the fact that translation is basically a process of equal substitution.
The point here is that there is no deep transcendental meaning inherent to the fact that .999... = 1.
.999... = 1 because it is forced to. It is a tautology that arises from the use the '=' sign and the infinitely repeating decimal notation, which are nothing more than convenient mathematical artifacts. And it only applies when limited precisely to mathematical equations.
Ultimately it is just a rule of the mathematical game. It is basically the same as the 'out of the pocket grounding' rule is in football.
Of course, if you want to play the game, you should know the rules. But you should not be under the impression that this is Divine Law.
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