From 1 to ... Graham's Number??? (Number talk 1)

     Let's get started... with this week's number talk! Yep, that's right, today, we'll be talking all about numbers. Okay, get your thinking caps on, and we'll start.

Numbers:

1 million (10 to the power of 6)

10 million (10 to the power of 7):

This brings us to the number of steps it would take to walk around the earth, 40 million steps. Still pretty small and imaginable (wait until we get to Graham's number)

100 million (10 to the power of 8);

Finally, we get to the number of books ever written in human history (130 million) and the average amount of words a human speaks in a lifetime (860 million).

And... this is getting boring, but for more, visit this site.

Now for a googol, which is 1 followed by 100 zeroes, which is also 10 to the power of 100.

And some more:

10 to the power of 113 is the number of hydrogen atoms that would fit in the universe when packed tightly.

10 to the power of 122 is the number of protons you could fit in the universe.

And let's go bigger, for a googolplex. 

     A googolplex is 10 to the power of googol. That's right, you heard me correctly. 10 to the POWER OF A GOOGOL! Holy Moly. If you want to write all the zeroes in a googol, first, buy a microscopic pen. Then, write 10 BILLION zeroes on EACH SINGLE GRAIN OF SAND ON EARTH. If I dedicated 80 years to writing zeroes, 3 zeroes per second, 365 or 366 days a year, 16 hours a day, I would still only have half a grain of sand done. If the whole human race that ever existed did this, we would still only have a cube with a side length of 1.7 meters full of sand. That's it.

     Yeah. So big, right? I hope you are completely mind blown by now. But if not... we are going to get to it. Graham's number will completely dwarf a googleplex. Oh, you'll see. But first, we're going to introduce some terminology and stages of operations. Pay CLOSE ATTENTION.

Level 0: Counting
Yeah, I know, I know. You start from 1 and count until you get to the number you want. Not efficient.
Level 1: Addition
Much better than Counting.
Level 2: Multiplication
Starting from here, numbers start to get big quickly. Multiplication is also iterated addition.
Level 3: Exponentiation
Ah. Things start to get complicated. Exponentiation can be written as one up arrow, like this: ↑. Also, exponentiation is repeated multiplication. For example, 2 ↑ 3 = 2*2*2 = 8. 3 ↑ 4 = 3*3*3*3 = 81. Make sure you understand this before you move on.
Level 4: Tetration
Okay. Things are now completely mind-boggling and you will probably not hear the word "tetration" many times. However, you probably guessed it by now, tetration is repeated exponentiation and is denoted as ↑↑. For example, 3↑↑3 = 3↑3↑3 = 3↑27 = 7625597484987. And also 2↑↑4 = 2↑2↑2↑2 = 2↑16 = 65536. So, as you can see, things are getting pretty big. But yeah. We have two operations to go.
Level 5: Pentation
Oh my gosh. Yep, pentation is repeated tetration and is denoted as ↑↑↑. For example, 2↑↑↑3 = 2↑↑(2↑↑2) = 2↑↑4 = 2↑2↑2↑2 = 65536. Yes, I know. So small, you say. Let's try something bigger. 3↑↑↑4 = 3↑↑(3↑↑(3↑↑(3↑↑3))) = 3↑↑(3↑↑(3↑↑(3↑3↑3) = number that's too big to write. Yeah, things are growing, really fast.
Level 6: Hexation
By now, your brain is probably telling you to stop. But no. We have a lot to go. Hexation is denoted as ↑↑↑↑ and is repeated Pentation. And I think you get how it works. Even 3↑↑↑↑4 is a gigantic number, too big to write down and even imagine.


Okay, now that we are done with the terminology, time to teach Graham's number. Set g1 to 3↑↑↑↑3. Not so big yet, right? Yeah. Agreed. Well now, things are about to get scary. g2 = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑...↑↑↑↑↑↑↑↑↑↑↑3. So, you say, how many arrows are there? The answer is g1. Yep. You heard me right. THERE ARE g1 ARROWS IN g2. But we are not done. g3 = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑......↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3. And yep, there are g2 number of arrows. But g3 still isn't Graham's number. Neither is g4 or 5 or 6 or 10 or 25 or 50 or 60. None of those are enough. Graham's number is g64. Yes. Note: I would have put five exclamations next to the g64 but I was afraid they would be confused as factorials. Anyways, yeah. Graham's number is g64. Too big to imagine. But... there's more. There are numbers that dwarf Graham's number. I'll explain those numbers if this post gets 300 views.

Have a nice day, and I hope you enjoyed this number talk.