How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out". I use an "expand-o-tron" analogy to wrap my head around what exponents are really doing: some amount of growth (base) for some amount of time (power). This gives you a "multiplier effect". So, 3^0 means "3x growth for 0 seconds" which, being 0 seconds, changes nothing -- the multiplier is 1. "0x growth for 0 seconds" is also 1, since it was never applied. "0x growth for .00001 seconds" is 0, since a miniscule amount of obliteration still obliterates you.
It was a fascinating discussion, but I'm most interested in the bolded part which says it perfectly - "a minuscule amount of obliteration still obliterates you."
This is very important in business, especially if you're doing online type stuff where many of the probabilities are invisible to you unless you carefully study the analytics. Analytics-inclined people already do this, but in 2011 that's still definitely a minority of the business population.
So if you want to get a sale, you might have something like this -
You send out 10,000 flyers.
9,000 go in the trash without being looked at. 1,000 are glanced at.
Of those, 100 people actually go check out what you're offering.
50 of them like it and decide to buy.
40 of them can figure out the "Checkout" button on your website (seriously).
30 of them successfully put in payment options and pay.
10% * 10% * 50% * 80% * 75% = .003 = .3%
30 buyers out of 10,000 people solicited.
Now, numbers are pretty amazing. The first and most crucial lesson here is plugging a 0 anywhere into the equation drops your success down to 0.
It doesn't matter where - if you put a *0% anywhere (your payment options are broken, your website URL is broken, you forget to mention your website on the advertisement - don't laugh because it actually does happen) - then you lose.
So, you might be laughing at this point. What kind of idiots do that? Well, the most common *0% isn't technical errors which eventually get sorted out, but rather not trying a particular marketing channel at all. You're getting 0 sales and 0 profits out of anything that you're not trying at all.
Unfortunately, when you do try a new marketing channel, that's when you don't understand the benchmarks of what's possible and what's common. If you're multiplying by 0 or otherwise a very low number because you made an error with your marketing copy or your offer or your technical side of things wasn't correct, then the marketing channel looks like it sucks and you move on. But it might have been worthwhile and useful.
If you were getting a magazine advertisement, for instance, you might find that:
"TRY NOW FOR FREE: productnamefreetrial.com" might very much outperform "Find out more" or something more generic. Of course, many people write advertisements that sound reasonable and unobtrusive and don't want to be obnoxious, so they write "Find out more" in their ad, of the 50,000 readers of the magazine they only get 20 visits to their website, and they conclude that advertising in that magazine sucks and doesn't work.
But it's multiplying by a very low number - not quite obliteration (which entirely kills you at any point in the process) but something near obliteration wrecks your numbers.
How to deal with it? Find someone in a non-competing industry using a particular channel successfully and knowledge share with them on what benchmarks are common, and work and fight until you get near those benchmarks, or it appears like you can't get near those benchmarks with your particular product/offer/channel mix.
Of course, that takes a while and can be a big stressful pain in the ass, which is why most small business owners don't venture too far away from their established channels after a while and thus give up a lot of growth - after all, the most common form of multiply-by-zero is not trying.