Graham's Number

For this exploration, I have decided to focus my research on a subject I find quite interesting and intriguing, and that topic is Graham’s number. The reason I find this topic to be so fascinating is because it’s a very large number. Quite literally. Its size is less than infinity, but the number itself is so large, that if a person tried to imagine it in his/her head, their head would collapse on itself and form a black hole. This is actually not a hyperbole, it’s a fact. It is hard to believe, but it’s a fact. This number is so huge, that if all matter in the universe becomes paper and ink, it still wouldn’t be enough to write all of the number down. That’s why I like this topic (Graham’s Number, Numberphile).

Ronald Lewis Graham is an American mathematician who comes from California, and who was born in 1935. His work revolves around looking for patterns in chaotic systems, and so he pursued a pattern involving joining up four points with six lines, all red or all blue, in a tube of many dimensions. This pattern is repeated many times in an all-blue cube, but Graham changed the color of individual edges, trying to avoid the pattern in just one color. His aim was to see if avoidance of the pattern in blue would force it to pop up in red. This doesn't happen in three, four, or even five dimensions. The number of dimensions required to guarantee this pattern turns out to be Graham's number. This part of mathematics is called Combinatorics, and Graham started looking into a more specific field of it, called Ramsey’s Theory. This theory could be explained the following way; this is an example of where complete disorder is impossible. In any large system, you've got to have a smaller set that has a lot of structure to it....

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...for modelling large management and scheduling problems. These include the optimization of flight schedules, the scheduling of staff, and the planning of equipment maintenance. Combinatorics is also used in areas as miscellaneous as cryptography (which is a necessity for top secret government issues, among other government-related matters, as to prevent hacking), planning an efficient layout of a factory floor, electronic security, improving telecommunications signals, and synchronizing traffic lights. One could clearly see the major effect this part of mathematics has on our modern, advanced world. It really is an important topic that is quite often overlooked at by many. Some job titles it offers are: Software Developer; Research analyst; Consultant; Application developer; and Quality assurance analyst. (Combinatorics and Optimization, The University of Waterloo).