Friday, February 10, 2006

The biggest number used in mathematics

Well, It is smaller than infinity, but it is larger than any number you have ever imagined.

It is the Graham's Number.

Let's first talk about a relativly small number, a Googol.

A Googol is larger then the age of universe in seconds almost 6 times larger, so you know you can't count a Googol, although a very small number. It is actually equivalent to 10100, e.g. 1 and 100 zeroes next to it.

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A googol is greater than the number of particles in the known universe, which has been variously estimated from 1072 up to 1087.

If you drew a regular polygon with a googol sides that was 1027 times the size of the known universe, it would still appear circular, even on the scale of a Planck length. Planck length is smaller than 10^-39 of a meter.

There is Googolplex which is 1 and a googol zeroes next to it. And a Googolduplex which is 1 and a goololplex zeroes next to it. And there is a Googoltriplex which is 1 and a googolduplex zeroes next to it. There is also a Googolquadriplex which is 1 and a googoltriplex zeroes next to it.

The smallest one, googolplex, to write it under the following conditions: a printed page can hold 100 rows of 100 chars. 104 characters. Suppose we have a billion (109) of printers, each one printing a billion of pages each second. Then, after one year we could have printed 104 × 109 × 109 × 365×24×60×60 = 10×22 × 31536000 ~= 3.2 × 10×29 digits. After one billion years of printing, we'll have about 3.2 × 1038 digits. We still have 10(Googol-38) digits to write.

Thinking of this another way, consider printing the digits of a googolplex in 1-point font. 1pt font is .3 mm per digit, which means it would take about 3.5 × 1096 meters to write it in one point font. The known universe is estimated at 7.4 × 1026 meters in diameter, which means the distance to write the digits would be about 4.7 × 1069 times the diameter of the known universe. Regardless of the time needed to write it in the first place. Imagine how many light years you have to travel to write it. It would be larger than a billion (much larger) years if you are travelling at light speed.

That was a googolplex. So what about a googolquadriplex ? Seems a funny question. Actually the Graham's number is so larger than a googolquardiplex. It, as a matter of fact, can't be written using standart “exponentiation'' convention, like ab. It needs a special notation so that we can express it. If we used normal conventions, it would take more than it takes to write a googolplex, I assume.

Graham's number is related to a problem in a branch of mathematics called 'Ramsey theory'. Here's the problem:
Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph using only the colors red and black. What is the smallest value of n for which every possible such coloring must necessarily contain a single-colored complete sub-graph with 4 vertices that lies in a plane?
Although the solution to this problem is not yet known, Graham's number is the smallest known upper bound.
Martin Gardner wrote “Ramsey-theory experts believe the actual Ramsey number for this problem is probably 6”, making Graham's number perhaps the worst smallest-upper-bound ever discovered.

Sources :
http://en.wikipedia.org/wiki/Graham_number
http://en.wikipedia.org/wiki/Googol#Trivia
http://en.wikipedia.org/wiki/Other_names_of_large_numbers
http://en.wikipedia.org/wiki/Moser%27s_number
http://www-users.cs.york.ac.uk/~susan/cyc/g/graham.htm
http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
http://en.wikipedia.org/wiki/Hyper_operator
http://en.wikipedia.org/wiki/Conway_chained_arrow_notation
http://en.wikipedia.org/wiki/Skewes%27_number
http://en.wikipedia.org/wiki/Large_numbers#Approximate_arithmetic_for_very_large_numbers