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# Legally Binding Use of Factorials

You saved *how* much?
 (+54, -2) [vote for, against]

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!"

 — dryman, Jan 31 2004

Factorial Tutorial http://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

superfactorial http://mathworld.wo...Superfactorial.html
[yabba do yabba dabba, Oct 04 2004, last modified Oct 21 2004]

Wikipedia on the Billion http://en.wikipedia.org/wiki/Billion

dry++
 — theircompetitor, Jan 31 2004

Heh!
 — egbert, Jan 31 2004

Top Quality!. In this world of litigation we should stand up to people who deliberately mis-sell. I can't believe that I missed this.
 — gnomethang, Jan 31 2004

saying nothing....
 — po, Jan 31 2004

I predict that shortly this will become a top 10! idea
 — theircompetitor, Jan 31 2004

I predict you are wrong! !!
 — po, Jan 31 2004

Well, I'm not going to bet you 20!
 — theircompetitor, Jan 31 2004

Thanks for the croissant it's delicious! I am now trying to work out a way to use trigonometry to sue the manufacturer of a tan-colored cot.
 — dryman, Jan 31 2004

Now that is just a bad joke, dryman!
OH!SAH!COAT!
 — gnomethang, Jan 31 2004

Welcome to the bakery, dryman. (WTAGIPBAN)
 — krelnik, Feb 01 2004

What a fantastically obscure and pointless idea. +

+ for math humour, a genre often overlooked.
 — lintkeeper2, Feb 01 2004

+ Excellent. Re the [dryman] lawsuit, why not just cosh him over the exponent till he's powerless?
 — phlogiston, Feb 01 2004

Nah, that would be a sin.. ;-)
 — dryman, Feb 01 2004

 // And for reasons I'm not totally clear on, zero factorial is defined as being equal to 1. //

Convenience. (These are mathematicians we're talking about - what other reason could there be?)
 — Detly, Feb 02 2004

Nice. Especially the consideration of 0!

[Reminds me of my favourite promotional text which I saw in a newspaper ad: "Buy 3 for the price of 4 and get the 4th one absolutely free!!!"]
 — hippo, Feb 02 2004

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...
 — cFish, Feb 02 2004

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) ?
 — fusty, Feb 02 2004

according to my calculator, dryman is approximately at 4.3622!
 — theircompetitor, Feb 02 2004

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.
 — GenYus, Feb 02 2004

On a similar note, never go on any diet that promises "lose 10-20 pounds" -- you'll end up gaining weight.
 — phundug, Feb 02 2004

Very good. Sadly, the highest number of positive votes I can give you is 0!
 — AO, Feb 02 2004

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)
 — hippo, Feb 02 2004

 I always thought it was literarily exploded with rage.

Rage, rage against the dying of the light.
 — theircompetitor, Feb 02 2004

You get +!... how much is that in croissants?

 po, looks like I should have bet you that 20!

jutta -- how often does a first idea get this kind of reception?
 — theircompetitor, Feb 02 2004

Simply genious.....
 — KLRico, Feb 02 2004

[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.
 — Worldgineer, Feb 02 2004

Are you kidding me? What's anything divided zero times? It doesn't make any sense. Lim x--> 0 from both sides have to match.
 — yabba do yabba dabba, Feb 02 2004

We are agreeing.
 — Worldgineer, Feb 02 2004

 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!
 — dryman, Feb 02 2004

 Given that 1\$ = 1! = 1, and 2\$ = 2!^(2!) = 4,

I conclude that superfactorials are just the same thing as squares.
 — phundug, Feb 02 2004

 3! ^ 3! = 6 ^ 6 = 46,656, but I'm not so sure that's 3\$. Check the link--it looks like there's 2 different ways to do them.

3\$ = 6^6^6^6^6^6 = good luck., or 3\$ = 3! * 2! * 1! = 12
 — yabba do yabba dabba, Feb 02 2004

 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.
 — cFish, Feb 03 2004

you guys say a thousand million?
 — yabba do yabba dabba, Feb 03 2004

Yep, a Thousand Million is 10^9, and a Billion is 10^12... I've added alink to Wikipedia that talks about it..
 — cFish, Feb 04 2004

I'd pay good money to see someone literally explode with rage.
 — Size_Mick, Mar 07 2004

 // 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)
 — Ossalisc, Mar 25 2004

nice +
 — Lacus Trasumenus, Mar 30 2004

I absolutely love this and think we should all try to take advantage of this. [+]
 — lunaras, May 23 2004

I will definitely use this!
 — bspollard, May 27 2004

...in the title.
 — yabba do yabba dabba, May 27 2004

Now this is just brilliant. I'm tempted to write to any company I see using this from now on...
 — weezy, Jun 06 2006

croissant!
 — neilp, Jun 06 2006

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.
 — apocalyps956, Jul 20 2006

 //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?
 — MikeOliver, Jul 21 2006

[+]!^2
 — monojohnny, Jul 21 2006

Is that The Pope, [monojohnny]?
 — MikeOliver, Jul 21 2006

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.
 — kevinthenerd, Jun 10 2008

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