Millions in a trillion?

This video clip reminds me of an anecdote related in The Universe and the Teacup:

[A professor] likes to impress his students with the power of large numbers by drawing a line designating zero at one end of the blackboard and another marking a trillion on the far side.  Then he asks a volunteer to draw a line where a billion would fall.  Most people put it about a third of the way between zero and a trillion, he says.  Actually, it falls very near the chalk line that marks zero.

How Many Millions are in a Trillion? from Econ4U on Vimeo.

Subway Factorials

I was ordering a sandwich at Subway the other day when the checker said "Awesome shirt!  What are the exclamation marks for?"

I was wearing a shirt with one of Ramanujan's incredible formulas for 1/pi:



I was able to explain to him what a factorial is (6! = 6 * 5 * 4 * 3 * 2 * 1 = 720) but ran out of time before getting to double-factorials:



So, 6!! = 6 * 4 * 2 = 48, while 7!! = 7 * 5 * 3 * 1 = 105.  While double factorials look more impressive, they are always less than or equal to normal factorials.

I learned that double factorials are one case of multifactorials, where the number of exclamation points indicates the difference between factors:

8!!! = 8 * 5 * 2 = 80

10!!!! = 10 * 6 * 2 = 120

Looking at the entries for "factorial" on Wikipedia and Wolfram, I was surprised at how little I know about them.  I did find a few cool facts to take away, though:

• the factorial function growth faster than any polynomial or exponential function, but it can be approximated by an expression involving e and pi:
  


• the gamma function is just like a factorial but can take real and complex arguments (not just integers).  The gamma function does not return 0! = 1, so it must be defined as a special case.
  


• 0.5! = sqrt(pi) / 2

What other stuff is cool about factorials?

Population graph

This could be illuminating for a class to generate: a map of the world where area is determined by population.  Can you find Australia?  Doing the US would be fun, with a huge California and tiny Alaska.



Found at http://www.geohive.com/earth/gen_popsize.aspx

The Wooden Ratio

You've heard about the Golden Ratio, 1.618..., computed as the mean of 1 and sqrt(5), or as the limit of the ratios of successive terms in the Fibonacci Sequence.  This ratio rears its head in the natural world, music, art, and in the proportions within our own bodies.

I have recently become aware of another ratio with similar properties.  Computed as the mean of 1 and 3, or as the ratio of successive terms in the sequence {2^n}, I have dubbed this number the Wooden Ratio (after the two-by-four).  It begins 2.000... and we can represent it with the symbol þ (thorn).

While the Golden Ratio appears in architecture such as the Parthenon, just think about how many Wooden Ratios appear in all of the brick buildings made by humanity.

In each of these bricks, the length is þ times the width, and the width is þ times the thickness.