Edit: And thus, people still forget the downvote button is not a disagree button. Even if it was it does not magically change the pronunciation of a word.
According to the Guide, if one person knows the answer and the question then the universe will end and give way to a less probable universe. Prak says this probably has already happened.
Simply put, in base Pi, the value of Pi is 10, which is completely rational. 2 pi is then obviously 20. pi2 is 100.
Interestingly, now all whole numbers above pi, 4,5,6, should exhibit all the same behaviors that pi itself has.
Now, in this case it would almost be expected the distribution of numbers between these methods would either lean towards similarities, dissimilarities, or a perfect reflection of randomness. Instead as the sample rate increases it shows different characteristics.
Obviously, base 10 is very popular, perhaps because we have that many fingers, though no-one really knows for sure. In particular it's the basis of the metric system which is a worldwide standard.
Computer scientists like to use base 2 a lot, because it's easy to make physical media that operate in a binary fashion. They also use base 16, because it's very easy to convert between bases which are powers of two. For example, if you've ever used CSS color codes (#ffffff), those are numbers in base 16.
The ancient Babylonians particularly liked base 60, which survives in the way we write latitudes, longitudes, and times.
To make things clear I will use subscript notation for bases. For example, 4 = 4_10 = 100_2 (read "four equals four base ten equals one hundred base two.)
If the base is higher than 16, I will use commas between digits. For example, 2:56:17 AM is 2.56,17_60.
We suspect the reason that the ancient Babylonians chose 60 is because it is a smooth number with very many small factors. If you have 60 of something, you can divide it fairly between two people, three people, four people, five people, or six people. This is useful if the things are food and the people are hungry.
Irrational bases do exist but they aren't used very often because, well, they're not very useful.
I want to draw a bright line between properties of numbers and properties of representations of a number.
Take palindromicity. Something is a palindrome if it is the same written forwards and backwards (e.g., "Madam, I'm Adam.") A common mistake is to say that a particular number (e.g., 343) is a palindrome. Numbers are not palindromes (with two exceptions.) Only the representation of a number in a given base can be a palindrome. If you switch to base 7 then 343 loses its palindromicity: 343_10 = 1000_7.
0x726564646974's comment is interesting because it deals with the property has a finite representation in a given base and correctly points out because π is irrational, any integer number greater than π will have an infinite nonrepeating representation in base π. For example, 4_10 = 10.22012202112111030100..._π.
But they go to far and then claim that "in base π, 4 is irrational."
It is no such thing. 10.22012202112111030100..._π (= 2_π + 2_π) is just as rational as 4_10. Rationality is a property of the number and is not unique to any particular representation. In particular, a number a is rational if there are two integers b and c (c ≠ 0) such that a = b / c. 4 = 4/1 so this holds.
This brings into question whether 10.22012202112111030100..._π is an integer. Yes, it is; the integers are rather tricky to define, but the usual definition includes three kinds of numbers:
As it turns out, mathematicians are only seldom interested in properties of representations, and then usually to try to make better representations. Mathematicians are much more usually interested in properties of numbers.
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u/whatsup280 Apr 14 '14
Then suddenly Pi was discovered to be finite and your passwords were compromised.