When you feel down…

I’m sorry for not posting for a while. Online learning isn’t exactly the best. Everything’s been so pressurising on me, and I’m sorry about that.

But I decided to come back, and I’m writing again! (Yayyyyy)

Do you remember the Up-Arrow Notation? If you don’t I briefly explained it this post. I will go into detail in the future, but today, I will do the opposite.

Down-Arrow Notation!

There are three attributes to this notation, and it is:

a↓ⁿb = …

1. IF n = 1, a^b
2. IF n > 1 and b = 1, a
3. IF n > 1, b > 1, (a↓ⁿ(b-1))↓^n-1a

Confused? Me too.

Since lists don’t make sense at all, I’ll use words.

For n = 1, it’s just a↓b, and that is a^b. For b = 1, it’s just a. For other cases, it’s… whatever that is, or (a ↓ ⁿ(b-1)) ↓ ^n-1 a, so to calculate Down-Arrow Notation, you will have to go through a fair bit of recursion.

Now, side note, I don’t remember if I mentioned this, but ↓ⁿ means n arrows, so ↓⁴ is ↓↓↓↓.

Compared to Up-Arrow Notation, the Down-Arrow Notation is pretty weak, like this:

a↓↓↓b ≈ a↑↑(b+1) or a^a^a^a^a…^a with b+1 a’s

a↓⁴b ≈ a↑↑(a(b-1) + 1)

And also, for comparison, 3↓↓↓3 is 3³²⁰, while 3↑↑↑3 = 3^3^3^3^…^3^3^3 with 7,625,597,484,987 3’s.

3^3 = 27, so already, 3↑↑4 is larger than 3↓↓↓3, which implies that ↓ is very weak, and ↑ is very fast-growing. For very large numbers, both are compatible, but for numbers like Graham’s Number, the Up-Arrow Notation is more optimal.

Next time, I’ll go more in-depth into Up-Arrow Notation. Stay anticipated, my friends.