.123456... = x + 2 x^2 + 3 x^3 + ... with x = 1/10.
Then you have
(x + 2 x^2 + 3 x^3 + ...) = (x + x^2 + x^3 + x^4 + ...) + (x^2 + x^3 + x^4 + x^5 + ...) + (x^3 + x^4 + x^5 + x^6 + ...)
(count the number of occurrences of each power of x^n on the right-hand side)
and from the sum of a geometric series the RHS is x/(1-x) + x^2/(1-x) + x^3/(1-x) + ..., which itself is a geometric series and works out to x/(1-x)^2. Then put in x = 1/10 to get 10/81.
Isn't it essentially the same thing, but less formal
0.1111... is just a notation for (x + x^2 + x^3 + x^4 + ...) with x = 1/10
1/9 = 0.1111... is a direct application of the x/(1-x) formula
The sum of 0.0111... + 0.00111... ... = 0.012345... part is the same as the "(x + 2 x^2 + 3 x^3 + ...) = (x + x^2 + x^3 + x^4 + ...) + (x^2 + x^3 + x^4 + x^5 + ...)" part (but divided by 10)
And 1/81 = 1/9 * 1/9 ... part is the x/(1-x)^2 result
I don't know who downvoted this, but it's correct.
The use of series is a little "sloppy", but x + 2 x^2 + 3 x^3 + ... has absolute uniform convergence when |x|<r<1, even more importantly that it's true even for complex numbers |z|<r<1.
The super nice property of complex analysis is that you can be almost ridiculously "sloppy" inside that open circle and the Conway book will tell you everything is ok.
[I'll post a similar proof, but mine use -1/10 and rounding, so mine is probably worse.]
If you set x = 0.123456..., then multiplying it by (10 - 1) gives 9x = 1.111111..., and multiplying it by (10 - 1) again gives 81x = 10, or x = 10/81. I’m not writing things formally here but that’s the rough idea, and you can do the same procedure with 0.987654... to get 80/81.
I, for one, find it very cool that two out of the five people named as authors are from my old university Chalmers University of Technology, in Gothenburg, Sweden. Chalmers have been very strong on functional programming for quite a while, being (or have been) home to many influential people. Apart from those named in this paper, I can name John Hughes (Haskell/QuickCheck), Thierry Coquand (Coq) and Ulf Norell (Agda) just from the top of my head.
Wow, let me be the first to say Thanks! I still use Miranda daily, both at my desktop computer at home and at my computer at work. Just chat, no fuzz.. Just the way I like it.
It seems that Miranda has forked into Miranda IM and Miranda NG, what are your thoughts about those two? Personally, I use Miranda NG because the Facebook plugin worked better in NG last time I compared the two.
I'm out of the loop. Not even sure what the differences are. But as long as people are still interested in developing it and using it, I think that's good. Hopefully the fork will rejoin sometime in the future.
"The story of the seizure of the machine by Balme and his shipmates was kept secret until the mid-1970s". I've always been intrigued by this fact. Does anyone know why this was kept a secret for so long?
"An estimated 100,000 Enigma machines were constructed. After the end of World War II, the Allies sold captured Enigma machines, still widely considered secure, to developing countries".
I'm sure they were quite happy to sell Enigma and also decrypt their communications for nearly another 30 years.
If the Enigma machines were operated properly, they could never be broken. Breaking them were a result of German lack of discipline and irresponsibility, not machine's weakness (maybe a simple inscription on the cover of machine in the bold letters would help though, machine designers just expected too much from their enlisted, barely literate operators).
Not never. One of the "features" of the Enigma was that a plaintext letter could never be encrypted to itself. An 'A' going in could come out as any other letter, except 'A'. German information security policies were generally pretty good. There were lapses, of course, and some of these were used to form cribs for a known plaintext attack on encrypted messages. But Enigma was/is not invulnerable.
People found a couple encrypted Enigma messages after World War II. Here is a note from a group of people using modern computers and brute force to decrypt them:
Just as during WWII, during WWI, the British had a very sophisticated interception and decryption program[1]. Its capabilities were also kept secret for long after the war and the Germans remained largely ignorant of how thoroughly their codes had been broken. I think if the work during WWI had been revealed earlier, the Germans would have been more careful and would have avoided repeating some of the mistakes of WWI. It was shown to be a wise policy after WWI, so I'm not surprised that they continued it after WWII, especially with the start of the Cold War.
I can't speak about the relative sophistication of British cryptanalysis during WWI, but Bletchley owed much of its sophistication to the Poles. The Polish Cipher Bureau was responsible for bringing mathematical and computational techniques into cryptanalysis while the French and the British were still largely employing linguistic analysis. Few realize that the Enigma was actually broken and reverse engineered by the Cipher Bureau before two new rotors were added by the Germans less than a year before the War, or that Turing's bombe was developed from Rejewski's bomba.
The Poles certainly got the ball rolling (see http://www.codesandciphers.org.uk/virtualbp/poles/poles.htm), but the organization at Bletchley Park took decryption and analysis of Enigma traffic to an industrial scale. By the end of war Bletchley employed around 10,000 people. See Alan Turing's and Gordon Welchman's biographies for more details.
That sounds like the 30 year rule in action. At the time it was done, it would have been extremely secret, and secrecy was not abandoned simply because the war was over.
It's noble and all, but I think that the same decision would be made even if only business aspects were taken into account. Imagine being "that one ISP where you cannot download movies". They would probably loose many current and potential customers by having that stamp, even among customers that do not illegally download any movies at all.
I'm Swedish myself and I have Bahnhof as my ISP. They are a company which builds much of their image on issues such as integrity and privacy, and that image is what got me to choose them over other options which were slightly(!) more expensive. So, all the sudden you have a company that gets a lot of appreciation from the public, and it's not like they are spending more money on their infrastructure just because they do everything they can to avoid tapping their customers' traffic.
This seems like a no-brainer decision, which is why I find it hard to understand why it's not like this everywhere.
> They would probably loose many current and potential customers by having that stamp, even among customers that do not illegally download any movies at all.
I think you are grossly overestimating the amount of thought the general public puts into which ISP it uses (when it has a choice).
Here's an example. The structures inside the meteorite were't proved to be life but if they had been alive they would have survived the trip from Mars to Earth.
We have pulled living bacteria out of sediment 100 million years old so 17 million years is nothing. More importantly this is just the time for this particular meteorite - people have done the calculations and some meteorites do the trip in around a year.
