Diary of a mad man
Dogsbody
Just a guy, read the about me page for more info :-) +Dan Benton
Posts by Dogsbody
Beyond Infinity
01 Jul 1997
I’ve always had a fascination with mathematics and this news article about numbers larger than infinity has always enthralled me, I used to have it pinned to my bedroom wall when I was a kid (insert scary comment about posters on walls here), Now that I don’t have a bedroom wall that I can pin things to I thought I would put it up here!!
How can you be sure that truly insoluble problems actually exist, and that they’re not just waiting for a clever solution to be found? It was the British mathematician and computer pioneer Alan Turing who established that there are some things that computers will never be able to do.
Turing based his discovery on an extraordinary piece of work by the 19th century German mathematician Georg Cantor, who discovered that there are numbers greater than infinity. Cantor’s proof can be stated in many different ways, but one simple form is this: imagine that you could write down a table showing the binary form of every integer from zero to infinity. Each number appears as an endless string of ones and zeros. Now, mark a diagonal across the table so that it joins the first bit of the first number, second bit of the second number and so on, all the way to infinity. Next, flip every one of those bits on the diagonal, changing one to zero and zero to one.
The diagonal string of binary bits you now have clearly represents a whole number. And because your original table included every whole number, it has to be one of those. Or does it? It can’t be the first number because it differs in the first bit. It can’t be the second, because the second bit is different. It can’t be the nth, because the nth bits don’t match. So you’ve just created a number that isn’t present in the list of all numbers. The only way out of this paradox is to accept that the number of integers is actually greater that infinity.
Cantor established that there was a distinction between infinity which is countable, and so-called transfinite numbers which are even bigger and can’t be counted. Transfinite numbers are important in many areas of mathematics and computer theory. Turing applied them in a particularly ingenious way. He was able to show that the number of possible algorithms, or computer programs, is infinite, but that the number of possible problems is transfinite. Hence, some problems can never be solved by a computer.
Before 0 0 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1
After 1 0 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1
The ultimate answer to the universe
09 Nov 1996
Being a huge Hitch Hiker’s Guide to the Galaxy fan this article has been used in many pub arguments! Currently the ultimate answer is 70!
AMERICAN astrophysicists will plunge thousands of fans of The Hitch Hiker’s Guide to the Galaxy into confusion today.
They say a claim this week by British scientists that the answer to the fundamental question of the universe is 42 is wrong – the real answer is 65.
Cambridge scientists caused a stir yesterday after they announced that their new technique for calculating the Hubble constant – the holy grail of cosmologists – had yielded the same answer as the one calculated by Deep Thought, the fictional computer in the novel, The Hitch Hiker’s Guide to the Galaxy.
Deep Thought took seven and a half million years to find that the answer to Life, the Universe and Everything was 42. The British scientists took only a few years to come to the same conclusion.
But Princeton University scientists have just submitted a paper to the journal Astrophysical Journal Letters, in which they describe a new technique that put the answer at 65.
The argument is fundamental to cosmology, The universe is expanding as a result of the Big Bang. Hubble’s Law states that galaxies are moving apart at a rate that increases with their distance.
The Hubble constant is the ratio of the speed at which the galaxy recedes, to its distance from us. Knowing this will enable astronomers to work backwards and put a date on the Big Bang and thus the age of the universe.
Wesley Colley, one of the American researchers, said yesterday that the American technique, which uses a method known as gravitational lensing, was a simple one and therefore particularly reliable. The Cambridge scientists have also used a new, but different method.
The dispute should go on for some time. Over the last two years different scientists have calculated the Hubble constant to be 50, 55, 57, 60 and 80.
By Aisling Irwin
NOVEMBER 9, 1996
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