I think it starts with the Babylonians and 360.

If you observe the night sky carefully, day after day (or should that be night after night ) you'll notice that the stars move on a bit each night. Without access to a digital watch, it's reasonable to define this amount as one degree, and a whole year's worth as 360 degrees. It's a little out but not bad for the time.

360 has lots of factors, including 60, 30, 24, and 12; so a time system based on those numbers makes a lot of sense. 30 days in a month, and 12 months in a year; yes, it all makes lots of sense. In fact, their astronomers were accurate enough in their observations to know that none of these numbers quite works, but many civilisations have based their philosophy and religion around the mystical power of certain numbers, so why spoil a good thing? If the 'ordinary' folk cannot get it right and the leaders can, that just enhances their power.

Exercise: If a year is 365 days, and assuming the Earth's orbit is a circle, how long does it take for the Earth to rotate exactly once on its axis? (The answer is not 24 hours)

Bob

]]>2. Let

Xbe a topological space. Then the empty set andXare sets that areboth open and closed!

Open and Closed Sets have got nothing to do with the day-to-day open and closed.

]]>Happened to come across this old thread via a google search. Anyways, we were asked to prove this for a homework assignment. Obviously Cantor's proof is elegant and so it is widely used. For my proof, I constructed a 1-to-1 mapping with the natural numbers mapping to their reciprocal. Then I merely pointed out a real number in that interval (I used 2/3).

Does this properly prove that there are more reals than natural numbers or am I missing something?

I don't know if n00b is still around, but yes, if I understand your description of your proof correctly, then you have missed the whole idea.

It is not sufficient to show that there is a 1-1 mapping from the natural numbers into the reals that misses some numbers. After all, your mapping, and your example, were actually in the rational numbers. So if your reasoning were correct, there would be more rational numbers than naturals, which is false. In fact, by the same idea, a simple mapping would show that there are more natural numbers than natural numbers! The 1-1 mapping

maps the natural numbers to themselves, but misses 1 (or if you prefer to define the Natural numbers to include 0, it misses 0).

What these mappings really show is that the natural numbers are infinite (the definition of "infinite set" is "a set which has a 1-1 mapping with a proper subset of itself"), and that the cardinality of the Reals and Rationals are both greater than or equal to the Naturals, which also follows from the simple fact that the Naturals are a subset of both.

By definition, two sets are the same size, or cardinality, if there is a 1-1 correspondence between them that includes every element of both sets. Cantor's proof shows that every mapping from Natural numbers into the Real numbers must miss at least one real number. Therefore the cardinality of the Real numbers cannot be equal to that of the Naturals. Combined with "greater than or equal to" already noted, we get that the cardinality of the Reals is strictly greater than that of the Naturals.

]]>That is tentative. I am not too good with vectors. Could be a load of kaboobly doo.

]]>y = |x^x|

Isn't it neat that the local minima of the doman -5<=x<=5 is e^[-1/e] at x = 1/e?

Here's a link to the calculation:

http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427evkrcm4invi