As most half-bakers no doubt are aware, placing an exclamation point (!) after a positive integer n signifies the factorial operator n!=n(n-1)(n-2)...1, meaning that all postive integers less than or equal to the given number should be multiplied together to arrive at a solution. For instance,
1!=1
2!=(2)(1)=2
3!=(3)(2)(1)=6
4!=(4)(3)(2)(1)=24

and so on.

The arithmetic is really pretty straightforward -- but inexplicably, print and internet advertisers in particular appear to have fallen away from proper mathematical usage when financial matters are involved. I can't tell you the number of times I've been shocked (shocked!) at a coupon or advertisement brazenly promising to "SAVE $5!", while clearly having no intention of coming across with the actual sum depicted ($5!, or $120).

It is time that they pay for their enthusiasm.

Should you be so fortunate as to receive a coupon, spam email, or fake promotional check with any amount of money followed by a !, immediately present yourself at corporate headquarters with legal and mathematical representation and demand the full value to which you are entitled. The actual savings due on a coupon for $10! would come to just over 3.6 million dollars, while a sweepstakes offer such as "Win $50!" could result in an arbitrarily large, effectively infinite cash settlement.

There are a few caveats: Frustratingly, multifactorials such as n!!! increase somewhat more slowly than simple factorials n! for all values of n, giving particularly obnoxious offenders ("in less than 5 minutes, you can make $1000!!!!!!") a very slight reprieve. A case could be made that currency units distribute within the expression; thus a ($3)! coupon might only be redeemable in cubic dollars -- but I feel sure the parentheses would have to be expressly included in the promotional materials for that to be true. And for reasons I'm not totally clear on, zero factorial is defined as being equal to 1. So a credit agreement where "Your first month's payment is $0!" would actually go into default if you neglected to send in the $1 check.

All kinds of written communication could benefit from a more literal appreciation of this overlooked operator.
If after having won a $20! cash giveaway, lawyers for the advertising firm contact you with an offer less than the gross national product of Brazil, simply respond with the following message:

"I'll get back to you in several days -- I'm not sure how many, but it definitely will be less than 10!"

Factorial Tutorialhttp://en.wikipedia.org/wiki/Factorial Plaintiff's Exhibit A, may it please the Court. [dryman, Oct 04 2004, last modified Oct 21 2004]

Behold!http://www.newdream...ld/numbers/fact.htm [po, Oct 04 2004, last modified Oct 21 2004]

Wikipedia on the Billionhttp://en.wikipedia.org/wiki/Billion 1,000,000,000,000 [cFish, Oct 04 2004, last modified Oct 21 2004]

125! years of automobileshttps://www.faceboo...0.340071.6604386669 Did you know that the first motor-car was built and run before this universe began? [pocmloc, Jun 17 2021]

Let's get algebraic.
What is x! ?
better still, what about lexicographic factorials?
How does dryman interpret a word followed by an exclamation mark?
Does dryman! become (dryma)(drym)(dry)(dr)(d) ?
Or do we work back through the dictionary, and get a string like.....
zenonian! = (zenith)(zenick)(zendic)........(a) ?

I tried a similar thing at an office supply store. They had folders on sale for .05¢ I tried to point out that that meant I could buy twenty folders for a penny, but the clerk wasn't smart enough to figure it out.

Misuse of the word "literally" is
always funny, and very common -
compare the fairly ordinary
metaphor "he exploded with rage"
with "he literally exploded with
rage".

([cFish] -
interestingly, your formula n!/n =
(n-1)! with n=0 implies that 1/0 =
(-1)!, or (-1)! is infinite)

[ydyd] Yes, I realized this and removed my anno. But shame on you for trusting your calculator. It can be easily found through logic. 1/x as x --> 0 from the right tends toward infinity. 1/x as x --> 0 from the left tends toward negative infinity.

Oh gosh n' golly, I may have just made us even richer..

Looking toward the bottom of my link, it seems that the "superfactorial" of n is denoted n$. Now if you can find a coupon for an amount of money with vertical symmetry (say, $8), then what a simple matter to turn it upside down.. and solve the expression 8$ to see how much you've saved.

I don't actually know that much about math, but it seems to me from the wikipedia that what you would do is raise 8! to the (8!)th power, then raise that solution to the (8!)th power.. and do this a total of 8! times. Is this correct? Now *that* is a blisteringly, mind-meltingly large number. And it applies to every $8, $11, (or even better, $66) coupon on the planet. Advertisers -- beware!

In the first case: given that 3$ has well over 10e10 digits [rough mental calculation, might be a bit off], I think it'd be safe to assume there are very few places that'd give change for your voucher...

PS. that's 10 with 10 zeros after it, I'm thinking that'll be thousand millions for us UK speakers and billions for those state-side, and that's only the number of digits.

// I think that 0! = 1 comes from the patern that n! / n = (n-1)! , thus substituting 1 for n gives 1!/1 = 0! Sorry slightly off topic...//

if 0! was found to be equal to zero early on there would be no need to define it as such... 0! is defined as zero because the factorial was first used to find the number of orders a set of n objects could be placed into, eg. {1, 2} could be ordered alternatively only as {2,1}... 2! = 2... since a set of zero objects can only be placed into 1 order(s) 0! was defined as 1... Later this definiton was found mathematically when the factorial was proven to be equal to the Gamma function...

It's intresting that the equality you described seams to work well even for negative numbers (I haven't checked them all but I've just glanced at a few tables)

If you applied this concept to the ads that say "you are our 1,000,000th visitor!" you would actually be aproximately their (8.26*10^5565708)th visitor.

//If you applied this concept to the ads that say "you are our 1,000,000th visitor!" you would actually be aproximately their (8.26*10^5565708)th visitor.//

Would it not need to say "You are our 1,000,000! th visitor" for this to apply?

I'm digging it. Our society is growing numb to the abuses of advertisers. Everything is blown out of proportion to get your attention. This would be a great way to both stop that and to encourage public numeracy.

I was trying to figure out how to squeeze Oklahoma!
into this idea. Perhaps inventing a series of
musicals called Alabama!, Alaska!, Arizona!, .. and
so on until Wyoming! And figuring out what the
product of a series of musicals might actually sound
like. It could be a big hit.