All Your Base

A second post in one day, you cry? Yes, your eyes tell you the truth. I have been lax of late and updates have been sparse. Need to catch up!

Numbers are not the symbols with which we represent them.
What a number is is a matter for philosophy and logic and I may return to it on a later date. But today I'm going to talk about the ways in which we represent these entities.
There are two basic ways you can represent a number. You can have an additional system, whereby each symbol stands for a certain amount and you can combine these symbols by addition to represent the number that they sum to. (I suppose if you gave each prime a symbol, you could also do it by multiplication) Of this type are Roman numerals where I represents one, V five and so on to give numbers of the form MMCCCLV. This particular system has alternative forms: IV instead of IIII which can cause confusion. But the main problem is that it is somewhat unwieldy.
A better system is place values. We come up with a few symbols to represent small numbers and then construct bigger numbers by multiplying by a common factor. Our decimal system is like this: 123 means 1x100, 2x10 and 3x1 all added together. The advantage of this is that it breaks up the number into pieces that fall in the same ratio as all other numbers. This makes sums easier to do, since we can add the components or whatever.
This system - ours we call Arabic numerals although they are Indian in origin - took some time to catch on in Europe because it needs a placeholder ie. something to represent 0 and distinguish between 100, 10 and 1, for example.
Of course, there's nothing special about ten. The Babylonians used base sixty - that is, they had symbols for the numbers one to fifty-nine and after that they moved once place to the left to express sixty as '10'. We still preserve a relic of this - that's why there are 60 minutes in an hour and 360 degrees in a circle.
Binary is base two so only has two symbols to give numbers like 11010 (which is twenty-six). Hexadecimal is often used in computing because it provides a convenient shortcut for converting between our conventional decimal system and the binary used by computers. The numerals are 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F and 0.
So why is the base ten system so common? Well, the obvious answer is that we have ten fingers and it's therefore natural to count in tens. In one African language, the word for six literally means 'skip' ie. to skip to the next hand.
This is why when we send out messages into space in the hope of reaching an audience, we never do it in decimal. The odds of any listeners using decimal is low and they may therefore miss the message. Binary is a better choice, since it's the smallest possible place value system and is therefore the easiest to programme machines with. They'll probably think of looking for communications encoded in binary.

The Monty-Hall Problem. Interestingly, when an article appeared in a newspaper (I think it was the New York Times) that mentioned this and how switching your choice makes you more likely to win, thousands of people wrote in to criticise the paper and its apparent inability to calculate odds. But the paper was right.