# Section 2 - Recursion and Common Notations

2.01. Introduction to Recursion

Before we can begin looking at the popular large number notations, we need to familiarize ourselves with *recursion*, which is arguably the single most important tool for generating large numbers. This short article explains what recursion is and why it is important to googology.

2.02. Knuth's Up-Arrows and the Hyper-Operators

Our first large number notation to examine is Donald Knuth's famous up-arrow notation, a way to generalize expressing the *hyper-operators*. It's foundational to various other large number notations.

2.03. The Weak Hyper-Operators

The weak hyper-operators are a less powerful variant of the usual hyper-operators. In this article we'll examine numbers that these operators make, explore the notations and compare the weak hyper-operators against the normal hyper-operators.

The Ackermann function is a famous two-argument function that has a very simple definition but produces values as big as Knuth's arrows do.

2.05. Googological Prefixes and Suffixes

Some naming systems for large numbers focus on adding prefixes and suffixes to existing numbers to make them much bigger. Here, we'll learn about Alistair Cockburn's fuga- family of prefixes and various other affixes used to name large numbers.

2.06. Joyce's Googol-Based Naming Systems

Continuing the topic of large number prefixes and suffixes, we examine a strange set of number naming systems based upon the name "googol" by a mysterious figure named *Andre Joyce*.

2.07. Steinhaus-Moser Notation

Steinhaus-Moser notation is another simple way to produce very large numbers, and the creators have coined some numbers with the notation: the mega, the megiston, and the Moser.

Perhaps the most infamous of all large numbers is *Graham's number*, a number that arose from a field of mathematics called Ramsey theory. In this article, I discuss not only the size of the number itself, but also the fascinating history behind Graham's number and two variants of the number. The main goal of this article is to clear up all misconceptions about Graham's number.

The most powerful of the "popular" large number notations is probably John Conway's chain arrow notation, an extension to up-arrow notation which easily surpasses Graham's number.

2.10. Extensions to Conway Chain Arrows

Some people, such as Peter Hurford, have further extended Conway chain arrows. Here we'll examine some of those extensions, and then I devise my own extension to chain arrows.

We finish section 2 with a review of what we've learned in this section, and a preview of what we'll learn about in section 3.