tetration

English

WOTD – 8 March 2025

Etymology

PIE word
*kʷetwóres

From tetra- (prefix meaning ‘four’) +‎ (itera)tion, from the fact that tetration is in fourth place after addition, multiplication, and exponentiation.^[1] The word was coined by the English mathematician Reuben Goodstein (1912–1985).

Pronunciation

(Received Pronunciation) IPA^(key): /tɛˈtɹeɪʃn̩/
Audio (Southern England): (file)
(General American) IPA^(key): /tɛˈtɹeɪʃən/
Rhymes: -eɪʃən
Hyphenation: tetr‧at‧ion

Noun

tetration (usually uncountable, plural tetrations)

Notation

The following notations are also used to indicate tetration:

$a\upuparrows b$
$a\rightarrow b\rightarrow 2$

(arithmetic) The arithmetic operator consisting of repeated exponentiation, by analogy with exponentiation being repeated multiplication and multiplication being repeated addition, $^{b}a$ denoting $a$ to the power of $a$ to the power of $\ldots$ to the power of $a$ , in which $a$ appears $b$ times.
Synonyms: exponential map, hyper4, hyperpower, power tower, superexponentiation
- 1982 May, Rudy Rucker, “All the Numbers”, in Infinity and the Mind: The Science and Philosophy of the Infinite, Boston, Mass.; Stuttgart, Baden-Württemberg: Birkhäuser, →ISBN, page 69:
  You don't ordinarily hear much about tetration because it is so powerful an operation that tetrating even very small numbers with each other produces inordinately large numbers. A tetration is worked out below.
  $^{4}2=2^{(2^{({2^{2}})})}$
        $=2^{({2^{4}})}$
        $=2^{16}$
        $=64,536$
- 1993, Alexander Humez, Nicholas Humez, Joseph Maguire, “One”, in Zero to Lazy Eight: The Romance of Numbers, New York, N.Y.: Touchstone, Simon & Schuster, →ISBN, page 39:
  Repeated exponentiation—called tetration—is so rare that a shorthand for it is hardly worth the trouble, appealing only to research mathematicians. Most professional engineers and scientists never encounter tetration, and the shorthand for it is seldom taught anywhere but in mathematics departments. But it doesn't stop there, and for the almost-unheard-of "multiple tetration" there is yet a stronger cousin operation, waiting in the wings.
- 2016, Erik Seligman, “Simple Surprises”, in Math Mutation Classics: Exploring Interesting, Fun and Weird Corners of Mathematics, New York, N.Y.: Apress, Springer Science+Business Media, →DOI, →ISBN, page 11:
  Tetration is usually symbolized with a number drawn to the upper left of another number, as opposed to the upper right used for exponents. Let's look at an example: what is 2 tetrated to the 4th? We would draw this as a 2 with a little 4 to the upper left side: $^{4}2$ . This is equivalent to $2^{2^{2^{2}}}$ . […] Needless to say, on positive whole numbers, tetration causes values to grow incredibly fast. This probably helps explain why it's not too practical in real life: after all, the simple operation of exponentiation allows us to concisely express the number of atoms in the universe, approximately $10^{80}$ , or a 1 with 80 zeros. […] But even without real-life applications, there are some mathematicians that find tetration a very interesting topic to study. On small numbers, it can have some bizarre properties. For example, get out a calculator and try calculating some tetrations of the square root of 2.