Tuesday, 14 January 2014

Notes on the History of the Factorial

I recently came across a nice blog from Paul Hartzer who blogs at Hero's Garden about Kramp's work with factorials. It prompted me to share my more general notes on the early history of factorials.

I have a curiosity about the etymology and history of mathematical terms as well, so I have included some notes on the etymology of factorial at the bottom.

In his book on The Art of Computer Progamming, Donald Knuth points to an example of the factorial (in particular 8!) in the Hebrew book of creation.
The first use of a multiplication of long strings of successive digits for a specific problem may have been by Euler in solving the questions of derangements. "The Game of Recontre (coincidence), also called the game of treize (thirteen), involves shuffling 13 numbered cards, then dealing them one at a time, counting aloud to 13. If the nth card is dealt when the player says the number 'n,' the dealer wins (this is known in combinatorics as a derangement of 13 objects.). Euler calculated the probability that the dealer will win.

It should be noted that this problem was solved earlier, by P.R. de Montmort, in 1713, though his work was unknown to Euler."
In an article entitled, "Calcul de la probabilité dans le jeu de rencontre" published in 1753, Euler wrote.
which is translated by Richard J. Pulskamp as "The number of cases \(1^. 2^. 3^. 4 \dotsb m\) being put for brevity =M." Cajori points out that this was probably not intended to be a general notation, but a temporary expedient.

In 1772 A T Vandermonde used [P]n to represent the product of the n factors p(p-1)(p-2)... (p-n+1). With such a notation [P]p would represent what we would now write as p!, but I can imagine this becoming, over time, just [p] (De Morgan would do just such a thing in his 1838 essays on probability). Vandermonde seems to have been the first to consider [p]0 (or 0!) and determined it was (as we now do) equal to one. Vandermonde's notation included a method for skipping numbers, so that [p/3]n would indicate p(p-3)(p-6)... (p-3(n-1)). It even allowed for negative exponents.

Vandermonde's symbol for [P]n would today represent what is generally called the "falling factorial." The common symbols seem to be [n]k or Donald Knuth's suggestion of \( n^{\underline{k}} \). Similar symbols exist for a "rising factorial", (n) (n+1) (n+2)...(n+k-1). Knuth's pleasing mnemonic version \( n^{\overline{k}} \) and (n)k which is common in working with hypergeometric series and is called the Pochammer symbol, although he never seemed to have used it for that, and used it for the combination of n things taken k at a time \( \binom{n}{k} \). I think either approach could be easily extended to using \ (n/s) as the base with the "s" representing the "skip rate". So (5/3)4 could represent the rising step factorial 5 * 8 * 11 * 14.

The word factorial is reported to be the creation of Louis François Antoine Arbogast (1759-1803). The symbol now commonly used for factorial seems to have been created by Christian Kramp in 1808 according to a note I found in Lectures on fundamental concepts of algebra and geometry (1911), by John Wesley Young with a note on "The growth of algebraic symbolism" by Ulysses Grant Mitchell. It was in the Note by Mitchell (pg 239) that I found the credit for the symbol to Kramp. Kramp had previously used the word "facultes" for the process, but deferred in favor of Arbogast's term instead. Here is a translation from Jeff Miller's page, "I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend". Both Kramp and Arbogast were working with sequences of products. (Kramp's more general notation allowed for "the product of the factors of an arithmetic progression, that is,\(a(a+r)(a+2r)\dotsb(a+nr−r)\), I use the notation \(a^{n|r}\) is well described in the post mentioned above by Hartzer)

In his Dictionary of Curious and Interesting Numbers,
David Wells tells the following story: "Augustus de Morgan ... was most upset when the " ! " made its way to England. He wrote:'Among the worst of barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! ... which gives their pages the appearance of expressing admiration that 2, 3, 4, etc should be found in mathematical results.'"

Another early symbol (shown below) was also used. Here is the discription of its origin from the web page of Jeff Miller,
An early factorial symbol, was suggested by Rev. Thomas Jarrett (1805-1882) in 1827. It occurs in a paper "On Algebraic Notation" that was printed in 1830 in the Transactions of the Cambridge Philosophical Society and it appears in 1831 in An Essay on Algebraic Development containing the Principal Expansions in Common Algebra, in the Differential and Integral Calculus and in the Calculus of Finite Differences (Cajori vol. 2, pages 69, 75).

I later found a copy of the 1830 paper on Google Books, and here is the way Jarrett presented the notation:

The symbol persisted and both symbols were in use for some time. Cajori suggests that the Jarrett |n symbol was little used until picked up by I. Toddhunter in his texts around 1860, and it was the use of his texts in America that may have influenced its use in the USA where it was more popular than the current symbol until around WWI.
The image below is from the 1889 textbook, A College Algebra by J.M. Taylor of Colgate.

A second image shows that the symbol was still in use even after the textbooks had adopted the "n!" symbol. This image is a note on the top a page on combinations in the 1922 text College Algebra by Walter Burton Ford of the University of Michigan. The book uses the exclamation point notation, but the hand written reminder is in the notation of Jarrett (and perhaps the teacher of Ms. Mabel M Walker whose signature is in the front of the book).

I recently found even a later date of the use of the Jarrett symbol. In the Mathematics Teacher for February of 1946 the symbol is used in an article by C. V. Newsome and John F. Randolph in illustrating Newton's power series for Sin(x). The fact that it is done with no comment indicates it must have still been commonly used.
I also came across an Arabic use of a very similar symbol, that is apparently still current. A note from an AP calculus teacher in February of 2009 indicated that a transfer student from Egypt uses something like this symbol currently.

A variation of Vandermonde's [p/3]n which allow the symbol to be extended to the idea of multiplying every other number, or every third, etc. What is today called the double factorial, triple factorial etc.  The earliest use I can find of either the "!!" notation or the term double factorial is by B. E. Meserve, in 1948 (Double Factorials, American Mathematical Monthly, 55 (1948)) His usage indicates he is using a well understood term, and symbol so I suspect there is earlier usage.  For example the use of a double factorial, as in 7!! means multiply 7*5*3*1; and 7!!! would be 7*4*1 (every third multiple). This seems to be little or no improvement to my mind from the notation Vandermonde used for the same purpose. It is important not to confuse these symbols with (7!)! which is the factorial of 7! or 5040!.

I received a comment to this post from Maurizio Codogno who had an even later use of Jarrett's symbol for factorial. He writes, "I found the L notation for the factorial in the book The Math Entertainer,(by Philip Heafford) which is dated 1959 (I have the 1983 reprint) He even shared a digital copy from the book.
This may seem a big number of arrangements. It is the prod- uct of 6 x 5 x 4 x 3 x 2 x 1. Another way of writing this product is \( \lfloor6 \), or, as it is often printed, 6!. It is called factorial 6.
I am now wondering if the notation is still in use in some part of the globe.

A good approximation to n! for large values of n is given by Stirling's Formula, which probably ought to be named for De Moivre. \( n! \approx \sqrt{2\pi n} (\frac{n}{e})^n\)
The Factorial can also be generalized to the real and complex numbers using the Gamma Function 

There is also a subfactorial and symbol in math. I am still searching for links to early uses, variations, etc. What little I knew a few years ago (and today) is here. Would love to have your input.
Unfortunately the same symbol, !n, often used for the subfactorial, was applied in 1971 by D. Kurepa for the sum of factorials,\( !n=\sum _{k=0}^{n-1} k! \) so !5 would b 4! + 3! + 2! + 1! + 0! = 24 + 6 + 2 + 1 + 1=34 .
Amazingly these two seemingly unconnected sequences are related. For clarity if we call the subfactorial seqeunce S(n) and the factorial sum sequence F(n) then it can be shown that \( F(n) \equiv (-1)^{k-1} S(n-1)\) Mod n.

There are other variations on the factorial. The primorial is the product of all the primes less than or equal to n, and is usually expressed as n#, so 5# = 5*3*2. They are useful to prime hunters, and the term was created by the very successful prime finder, Harvey Dubner. I would love to have a source for it's use, or the creation of the symbol.

There is an alternating factorial which is the sum of the terms of a factorial sequence alternately added and subtracted. For example af(5) = 5!- 4! + 3! - 2! + 1!. The only symbol I have seen is af(n), but I think something like \( (n\pm)! \) would be somewhat elegant. Go forth and use it. Donald Knuth, are you reading this?

There is also a superfactorial, the product of the factorials from 1 to n, \(\prod\limits_{k=1}^n k! \).  I have seen the symbol of a heart suggested, so 3 (heart-shape) would be 1!*2!*31.

There is even a hyperfactorial, although I have never seen it in use. H(n) = \(\prod\limits_{k=1}^n k^k \) These get big in a hurry. (If you have information on the origin and uses of any of these, please advise.) 

The term factorial is drawn from the more common math (and English) term factor. The roots of both these words are in the word fact and its Latin root facere, to do. To know the facts, is to know what has been done. The person who does something is then called the factor. In business a factor was once a common term for one who buys or sells for another. Today the word agent is more common. Colonial businesses often employed a person to do various menial tasks, as a factotum, literally one who does everything (today we might call them a "gopher"). Things that were necessary in order to "do something" became factors in the event, and today you may hear a coach say, "Defense was the most important factor in our victory."
Factors then became the parts of the whole, and a factory was where they were put together to make a final product. These words run over into the mathematical meanings. The factors are the numbers that are put together (by multiplication) to make the product. Because the product is made up by putting together parts, it is called a composite number.
The word "measure" has often been used in much the same way we now use the word factor. In his Universal Arithmetick Newton distinguishes three kinds of numbers, "integer, fracted, and surd", and defines an integer as "what is measured by Unity." Frederick Emerson's North American Arithmetic(1850)says "One number is said to MEASURE another, when it divides it without leaving any remainder." (pg 18) Later it states," A number which divides two or more numbers without a remainder is called their COMMON MEASURE." This is after the definition of factor on page 12, and immediately precedes "A square number is the product of two equal factors" on page 19.

Other English words from the "to do" meaning of fact include facility (the ability to do), faction (a group working to do the same thing), facilitate (make easy to do) and faculty.
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