This-way-up and Knuth arrows

I was looking today at a cardboard box that had the “this way up” symbol on it and wondered whether there is a Unicode value for it.

ISO 7000 symbol 0623 This way up

Apparently not. But there is an ISO code for it: ISO 7000 symbol 0623. It’s an international standard symbol for indicating how to orient a package. The name says it all: this way up.

There is a similar symbol in math and computer science: ↑↑. This is so-called up-arrow notation, introduced by Donald Knuth in 1976 [1].

In Knuth’s notation, ↑ indicates exponentiation, i.e. repeated multiplication, and ↑↑ indicates repeated exponentiation. There’s a little ambiguity here: we have to clarify in what order we apply exponentiation. Knuth stipulated that ↑↑ is right-associative, i.e.

b \uparrow \uparrow n &=& b \underbrace{\uparrow(b \uparrow \cdots(b\uparrow b))}_{n \text{ copies of } b} = \underbrace{b^{b^{.\,^{.\,^{.\,^b}}}}}_{n \text{ copies of } b}

So, for example, 5 ↑↑ 3 equals 5125, not 1255.

In general, n arrows means to repeatedly apply n − 1 arrows. If we use ↑n as a shortcut for writing n up arrows, then we can define ↑n recursively as meaning we apply ↑n−1 n times.

What I find most interesting about Knuth’s notation is how rarely it is used. I don’t think it’s because anyone object’s to Knuth’s notation; it’s just that there isn’t much need for what the notation represents. It’s primary use may be theoretical computer science. There you sometimes want to construct functions that grow ridiculously fast, such as Ackermann’s function, and functions of the form an b are good for that.

This is curious. Multiplication is repeated addition, exponentiation is repeated multiplication, and so repeated exponentiation seems like a natural extension. I won’t say that it’s unnatural, but it is very uncommon.

Related posts

[1] Donald E. Knuth. “Mathematics and Computer Science: Coping with Finiteness”. Science. 194 (4271): 1235–1242.

4 thoughts on “This-way-up and Knuth arrows

  1. Have you read “the biggest number in the world” by Darling and Banerjee? If not it’s worth it! Knuth’s notation is just where they start …

    Long time lurker – thanks for all the interesting posts!

  2. I recall some video or blogpost mentioning this which called the operation tetration. The Numberphile video “Numbers too big to imagine” discusses tetration, but it’s only four months old and I first heard/read of tetration years ago. Apparently it’s not commonly known because the values go up ‘too fast to be useful’ whereas other not-as-quickly-increasing functions such as factorial and primorial have common uses in probability/statistics and number theory.

    It appears that many people rediscovered or reinvented it, as there are many notations for it:

  3. Perhaps it’s because is cooler?

    But more seriously, there may just be a sort of anthropic/selection effect on hyperoperations: because they grow *so* fast, there’s just nothing in physical reality that is described by more than a handful of possible hyperoperation expressions. (Think of it as a version of the pigeon hole principle: because hyperoperation notations allow compact expression of so many extremely large numbers, they therefore cannot compactly express many small numbers.) And while there are by definition that many in mathematical reality, most of them exceed our ability to say or prove anything interesting about them.

    You could also say that because sizes grow so fast, there has to be a cutoff *somewhere*, and up-arrows are just where that cutoff lands. If up-arrows were useful, then the next level up of up-arrows applying to up-arrows would then be the cutoff.

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