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:
- Euler's Formula:
|
eix = cos(x) + i sin(x)
|
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 [
e−e,
e1/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.
