Before we can understand how to bundle a string of exponentiation the way exponentiation bundles a string of multiplication, we need to understand what a “string of exponentiation” even is. All Rights Reserved. It’s really uncool to say the word quadrillion.3 Most people opt for “a million billion” instead. He could barely finish the question before Milton opened his un-nuanced mouth and declared the number googolplex, which he, in typical Milton form, described as “one, followed by writing zeroes until you get tired.”6 At this, Krasner showed some uncharacteristic restraint, ignoring Milton and giving the number an actual definition: 10googol or 1 with a googol zeros written after it. I can only wonder in awe what secret does TREE(4) and above holds! So what’s g1? 2 ↑ 3 = 8 And I think we’ve heard just about enough from octillion. If every single human dedicated every waking moment of their lives to writing zeros on grains of sand, as a species we’d have by now filled a cube with a side of 1.7m—about the height of a human—with completed sand grains. Written out, a googol looks like this: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. So what we have is: 3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333) = 3 ↑↑ (3 ↑↑ 7,625,597,484,987) = 3 ↑↑ (sun tower). So you know how upset I just got about this whole INSANITY thing? On my computer screen, that image was about 18cm x 450cm = .81 m2 in area. And again for g5. You’d need 1086 peas to make it happen. And just how long would it take to do that? Then that frenzy happens and produces an even more ridiculous number, which then becomes the number of towers for the next frenzy. The psycho festival ends when that final feeding frenzy produces it’s final number. The first tower (in blue) is a straightforward little one because the value of b is only 3: g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑↑ (3 ↑↑ 333). And then this happens again for g4. Addition is an iteration up from counting, which we can call “iterated counting”—so instead of doing 3, 4, 5, 6, 7, I can just say 3 + 4 and skip straight to 7. And that’s the number of arrows. So a googol is 1 with just 100 zeros after it, which is a number 10 billion times bigger than the grains of sand that would fill the universe. Double arrows. And we’ve learned that 333 = 7,625,597,484,987, so: g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑↑ (3 ↑↑ 333) = 3 ↑↑↑ (3 ↑↑ 7,625,597,484,987). Either way, there are about a quadrillion ants on Earth. 7,625,597,484,987), where the outcome of the first tower is the, (sun tower) is a feeding frenzy with a multiplied-out-sun-tower number of. Today, shit gets real. But what if we put two of these computations together, like: We get a power tower. Let’s call this tower the “sun tower,” because it stretches all the way to the sun. So: Now remember from before that 3 ↑↑↑ 3 is what turns into the sun tower. No one says quadrillion. Planck volumes in the universe is a joke. In fact, Graham’s number is practically equivalent to zero when compared to TREE(3). 10185 – Back to the Planck volume (the smallest volume I’ve ever heard discussed in science). 7,625,597,484,987) is our 150km-high sun tower: Why You Should Stop Caring What Other People Think. Ten: 10 (1 zero) Hundred: 100 (2 zeros) Thousand: 1000 (3 zeros) Ten thousand 10,000 (4 zeros) Hundred thousand 100,000 (5 zeros) Million 1,000,000 (6 zeros) Billion 1,000,000,000 (9 zeros) Trillion 1,000,000,000,000 (12 zeros) Quadrillion 1,000,000,000,000,000 (15 zeros) Quintillion 1,000,000,000,000,000,000 (18 zeros) Sextillion 1,000,000,000,000,000,000,000 (21 zeros) Septillion … As we’ve discussed, filling the universe with sand only gets you a ten billionth of the way to a googol, so what we’d have to do is fill the universe to the brim with sand, get a very tiny pen, and write 10 billion zeros on each grain of sand. We’re about to move up another level, and this is about to become more complex, so before we move on, make sure you really understand Level 4 and what ↑↑ means—just remember that a ↑↑ b is a power tower of a’s, b high. If each of your steps around the Earth were represented by a dot like those from the grids in the last post, the dots would fill a 6m x 6m square. I don’t think I’ve ever heard someone say “sextillion” out loud, and I hope to keep it that way. So many podcasts. And how high would that tower of 7 trillion-ish 3s be? People would be mad at you. And now that I’ve gained those tools (and you will too today), a googolplex to the googolplexth power sounds like a kid saying “100 plus 100!” when asked to say the biggest number he could think of. Okay so now let’s take a huge leap forward into a whole different territory—somewhere where the Earth’s volume is too tiny and the Big Bang too recent to use in examples. 1017 (100 quadrillion) – The number of seconds since the Big Bang. Comparing this to the bacteria fact, it’s like you have 1/10th of the world’s ants crawling around inside your body. That’s why we use powers. You can't express it in scientific notation, or even as a power tower. They spelled it wrong by accident.↩, When my father was my age, he had children.↩, Or a g65 number of years, which would be (3 [Graham’s number of arrows] 3)…or a gg64 number of years…I could go on.↩. So anyway, I said above that I had been limited in the kind of number I could even imagine because I lacked the tools—so what are the tools we need to do this? Tetration is intense. 1011 (100 billion) – This is about the number of stars in the Milky Way and the number of galaxies in the observable universe (100-400 billion)—so if a computer listed one observable galaxy every second since Christ, it wouldn’t be anywhere close to finished currently. Instead of saying 3 + 3 + 3 + 3, multiplication allows us to bundle all of those addition steps into one higher-operation step and say 3 x 4. That was the outcome of a feeding frenzy with only two towers. Just going to a fifth arrow would have made my head explode, but the number of arrows in g2 isn’t five—it’s far, far more than the number of Planck volumes that could fit in the universe, far, far more than a googolplex, and far, far more than INSANITY.