04 Feb Big Numbers
Today, we are going to talk about big numbers – really really really big numbers. So, you have been taking math tuition for a while now. What is the biggest number that you can think of that you have ever used? A million? A billion? A trillion? The numbers that we are going to talk about today are much bigger than probably what you have ever used.
Most of you probably have heard the story from your maths tuition teacher about how our favourite website “google” was named. For those of you who have not heard the story, the founders had the plan of naming it as “googol”. However, due to a spelling error, the website was registered as “google”. The word “googol” is not just any word – it is the name of a number. A very big number. “Googol” is the name given to the number which is 1 followed by 100 zeros i.e. 10100 in exponential form. How big is the number? The number of atoms in the universe is approximately 1080 – way too small when compared to “googol”.
“Googolplex” is a humungous number and the “googol” is just a tiny little blob when compared to this giant of a number. A “Googolplex” is 10(googol) i.e. 1010100. This number is enormous even when compared to “Googol”. It is a number which starts with the digit 1 followed by a “googol” zeros. However, googolplex is not the biggest number that we are going to discuss today. The size of the next number that we are going to discuss next is a sheer jaw dropper.
Mathematician Ronald Graham used this number in one of his papers and this number remains the biggest number which has been used in a published paper till date. How big is this number? This is much much much bigger than the “Googolplex”. Graham’s number is humongous; enormous. It is so big that it cannot be written in the normal exponential notation and an all new technique – known as the up-arrow notation is used to express this number. You have probably never heard your maths tuition teacher mentioned this number before. Who do you think may use these numbers?
TREE(3) and SCG(13)
Next we are going to discuss about TREE(3). The number TREE(3) was derived by Harvey Friedman and in terms of size, TREE(3) is even bigger than Graham’s number. Harvey Friedman also devised another large number known as SCG(13) which surpasses TREE(3) in size.
Loader’s number was derived by Ralph Loader. In 2001, a contest was held named Bignum Bakeoff . The objective of the contest was simple- in order to win, one had to write a C program in 512 characters or less producing the largest number. The program, of course, would be hypothetical as there is no computer capable of running such a program. A programmer named Ralph Loader came up with the C program which produced the Loader’s number – a humungous number. He subsequently went on to win the contest. The number that his program produces is much bigger than TREE(3) and Graham’s number. Only the maths tuition teacher who is a maths fanatic and LOVES maths will have an idea about these numbers.
Rayo’s number is named after Augstin Rayo and this number is known to be the largest named number so far. Rayo’s number is much bigger than any of the numbers mentioned above. In fact, the formal definition of the number says that Rayo’s number is “The smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less.”
Although these numbers are huge and their sizes are mind boggling, they are still less than infinity. Those who have taken some maths tuition on infinity and are familiar with the concept will be able realize the fact that although these numbers are quite big, they are no closer to infinity than much of the smaller familiar numbers with which we deal every day. So do you want to ask you maths tuition teacher if you will be learning these numbers?