History of Exponentiation

The story of tetration began with exponentiation.

Just shy of a score after the first records of the "+" and "−" notation appeared in 1526, in 1544 the German mathematician Michael Stifel published a detailed investigation of exponentiation in Arithmetica integra. Stifel discovered logarithms before John Napier, and gave an early form of logarithm tables. Stifel also coined the term exponent, also published in Arithmetica integra, but it did not find its way into the English language until 1704, when the term exponential appeared in the Lexicon Technicum by John Harris.

The Scottish mathematician John Napier is most remembered as the inventor of logarithms, although the idea had been around before him, he was the first to use the term, although it did not appear in English until 1615. Napier published his major work, Mirifici Logarithmorum Canonis Descriptio, in 1614, which contains several dozen pages of logarithm tables, along with a lengthy explanation of both logarithms and trigonometry. Napier's work made a huge impact on natural sciences across the board.

One of the most generous contributors to the development of exponentiation was Leonhard Euler (pronounced oiler). Euler coined the term function, and introduced the notation for trigonometric functions we still use today. Exponentiation had already been defined over the complex numbers, for some time, when he rediscovered in 1748 what is now known as Euler's Formula: LS( LI(Euler's Formula:, [ppE()ppSup(IU()ppV(x)) = cos(ppV(x)) + IU() sin(ppV(x))]) )TAB() It was later found, that Euler was not the first to discover the formula that now bears his name. It was first proved, in an obscure form, by Roger Cotes in 1714. Cotes is also known for working with Isaac Newton during the publication of his major work, Philosophiae Naturalis Principia Mathematica, commonly referred to as "The Principia."

History of Tetration

The product-logarithm, also known as the Lambert-W function, is indirectly related to tetration because it is related to the infinite iterated exponential function. (More detail can be found in Identities.) The product-logarithm was developed by French mathematician Johann Heinrich Lambert who published it in his first book in 1758. The infinite iterated exponential function is probably the first use of tetration, although the terminology has varied over time. It has also been called the hyperpower function, or simply referred to by its definition: "x^x^x^...". It was not until 1989 that Joseph F. MacDonnell used the term hyperpower for this function. In 1778, Euler was the first to mention this function when he proved that it converged on the interval eexe1/e. Another person to write about the infinite iterated exponential function was Eisenstein in 1844. Later, in 1925, Pólya and Szegö first used the "W(x)" notation that is commonly used today.

The notation yx is what most people use for tetration, and was introduced by Hans Maurer in 1901. Much later, in 1995, Rudy Rucker published a book called Infinity and the Mind in which he used this notation, and gave it so much publicity that Rucker is sometimes credited with its first use.

The Ackermann function is a topic of some confusion, because there are at least two versions of the function. One version has 2 arguments, and is closely related to hyper-operations. Another version has 3 arguments, and is equivalent to hyper-operations. Both of these functions are named after Wilhelm Ackermann who defined them in 1928. In 1953, Andrzej Grzegorczyk also published a paper on the hyper-operations, hence the sequence of hyper-operations are sometimes referred to as the Grzegorczyk hierarchy.

In 1947, Reuben Louis Goodstein coined the term tetration, and the names for other hyper-operations: tetration for hyper4, pentation for hyper5, hexation for hyper6, and so on, from the Greek number prefixes for the number that represents the hyper-operator. It is said that the second half: -ation, is short for iteration, but Goodstein himself gave no such indication.

The most modern analysis of tetration is Exponentials reiterated by Robert Arthur Knoebel published in 1981 in American Mathematical Monthly, a periodical also host to other tetration-related articles such as Infinite exponentials published in 1936 by Barrow about the infinite iterated exponential function. The most recent analysis of tetration is a website by Ioannis N. Galidakis called Extreme Mathematics. In it, Galidakis gives two extensions of tetration to real numbers, and an extension of the product-logarithm to quaternions.

In 1976 Donald E. Knuth invented a notation for hyper-operations called Up-arrow notation. It has been very successful on the internet, because it allows tetration to be represented as "x^^y". More recently, in 1996, John Horton Conway and Richard K. Guy published a book called The Book of Numbers in which they put forth yet another hyper-operation notation. Their notation, called Chained-arrow notation is much more expressive that Knuth's, and allows operations well beyond the hyper-operations. Another notation that goes beyond the hyper-operations yet still includes them was developed by Jonathan Bowers in 2002 called Array notation. Bowers' notation includes an infix notation he calls extended operator notation which is equivalent in all respects to the current de facto standard notation for hyper-operations, prominently used in Robert Munafo's website about Large Numbers.