theoretical physics
march 2015
BY andrew lee
Gargantuan Numbers
12
Prepare to have your mind blown.
When asked, “What is the largest number ever?” one would probably respond by saying “infinity” or “infinity + 1”. These answers are, in fact, flawed and are not sufficient to crown the largest number ever. In the physical world, we instead need to deal with numbers of consequence- not arbitrary checkpoints but quantities of real and practical value. So what is wrong with the number infinity? Let’s use the Hilbert’s paradox of the grand hotel. Think about a scenario in which we have a hotel with an infinite amount of rooms holding an infinite amount of people. If there was one customer who wanted to stay at the hotel, what would we do? We would tell the person in Room 1 to go to Room 2 and keep shifting customers up one room. When doing this, room 1 becomes open and the customer is satisfied. Thus, the following is true: =+1. We could also prove that =+. Therefore, we must look at finite numbers which are extremely large and have some sort of application/use
rather than infinities (refer to The Fault in our Stars for infinities larger than others). Before we understand extremely large numbers, we have to learn a new type of notation called Knuth’s up arrow notation.
Now we can start to express some pretty gargantuan numbers.
3 3
The depiction of these number is SO big that we cannot even comprehend how large it is. But there are some numbers that are even larger. Graham’s Number, for example, takes it even further. To understand Graham’s Number, we need to know the system of G’s which goes as follows: 33 =G1 3(G1)3=G2 (G1 number of arrows!!!) 3(G2)3 = G3 and so on…