Thinking Thomas

I have taken an interest in large numbers over the couple of months and the interest have accelerated over the last few weeks so I have compiled a list of interesting numbers for my own and others amusement. This was basically kicked off by me trying to calculate the third superfactorial. After a while I gave up. Today I discovered why I had to give up: This number has more than 10^((10^10)^10) digits! And to think that the first and second superfactorial is 1 and 4 respectively...

These are in increasing order of magnitude, some might in fact be equal, but I am not sure (this is the case for aleph0 and epsilon-null). The more obscure numbers should all be described in the links given at the bottom of this message.

• 0, nothing, just very special.
• i, the imaginary unit.
• 1, the unit. Not a composite and not a prime, so what is it? The same magnitide as i incidentally.
• 2, The only even prime. Also the first prime. Good enough to appear on this list :)
• e, pops up in every unimaginable situation.
• 3, A magical number.
• pi, pops up in every unimaginable situation. The cirumference diveded by the diameter. Brings us nicely to the beautiful equation e^(i*pi) - 1 = 0
• 7, another magical number. The seven wonders of the world, the seven seas and the seven dwarves. The icing on the cake is that it is a prime as well :)
• 42, the meaning of life, the universe and everything. Curtesy of Deep Thought and Douglas Adams.
• 666, the number of the beast. Speak to to John about this one.
• Googol, a 1 followed by 100 zeroes. Invented by someone who wanted a number larger than anyone would ever need (boy, was he wrong :)
• Centillion, The largest *illion number (not quite true really) 10^600
• Googolplex, A 1 followed by a googol zeroes. Invented by someone who wanted to prove googol wrong
• Mega, not as in 1 million but as in 2 in a pentagon, the googolplex is nothing compared to this
• The Moser, 2 in a megagon, on of the truly big boys, unimaginable large, the mega is nothing compared to this.
• Grahams number, the biggest big boy, unimaginable large, the brain gives up well before this point, the Moser is nothing compared to this number. It is my favourite number as well :)
• Omega, The first ordinal infinity. 1+ omega = omega, but omega + 1 > omega
• Aleph0, The first cardinal infinity. The number of integers. The size of an infinetely large object in euclidian space. Like an infinetely long line.
• Epsilon-null, the exponent tower of omegas with height omega. The first ordinal infintiy that cannot be expressed as a finite number of omegas using addition, multiplication and exponentiation.
• Aleph1, The first uncountable infitiy.
• The continuum, the number of points on a line, also the number of real numbers

Some more reading can be found here:
http://home.earthlink.net/~mrob/pub/math/largenum.html
http://www-users.cs.york.ac.uk/~susan/cyc/g/graham.htm
http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm