$2 \uparrow^n 2 = 4$ and the magnificence of $2$

Question

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because of the binary nature of binary operations. I require a formal way to prove that $2 \uparrow^n 2 = 4$
Please, can you help?

@user2345215: Oh I realized it just now! Dang, I was thinking in terms of tetration! — Nick, Mar 09 '14 at 12:21
@EricStucky: Sorry, I removed that part from my question. Just a logical error on my part. — Nick, Mar 09 '14 at 12:22
I removed the \huge because it was taking up too much space on the front page. — Dan Rust, Mar 09 '14 at 12:27
You were told to use induction. Where were you having trouble accomplishing that? Do you know what induction is? — anon, Mar 09 '14 at 12:39
@seaturtles: I see induction more as something that verifies something I have. I just hoped there was another way, ya know. — Nick, Mar 09 '14 at 14:23
@Nick: You're asking for a proof, which is exactly "something that verifies" the thing you want to prove. If you want something done that induction doesn't do, then proof certainly isn't what you want. — hmakholm left over Monica, Mar 09 '14 at 18:48

score 6 · Accepted Answer · answered Mar 09 '14 at 12:25

6

Well from the definition $a\uparrow^n2=a\uparrow^{n-1}a$ so for $a=2$ you get $2\uparrow^n2=2\uparrow^{n-1}2=\ldots=2\uparrow2=4$.

answered Mar 09 '14 at 12:25

user2345215

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It's silly how beautiful such a simple idea is to me. Indeed, it's very magnificent. – Nick Nov 16 '14 at 18:00

score 2 · Answer 2 · answered Mar 09 '14 at 12:24

2

Hint: Use induction.

If you have the luxury of defining your operation, you should define it recursively. If not, and your definition isn't recursive, then wow that's cool but you should probably tell us about it.

answered Mar 09 '14 at 12:24

Eric Stucky

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$2 \uparrow^n 2 = 4$ and the magnificence of $2$

2 Answers2