Here's a fairly silly example. At the store, the guy in front of you in line is buying 100 boxes of staples. Each one costs 90 cents. How much does the store get? The answer is 100 x (90 cents), or $90.00. This number is positive, which means crazy staple guy owes the store $90. In the rest of this story, positive amounts of money will always mean that staple guy has to give money to the store. Negative amounts of money, being the opposite of positive amounts of money, will mean the opposite -- the store has to give that much money to staple guy.
Whoops! Staple guy just remembered that he had a bag with 100 coupons good for 30 cents off of staples, and he wants to use them. The store has to return 30 cents per coupon to him, so they put a line on the receipt that says 100 x (-30 cents). This is recorded on the receipt as -$30.00. This number is negative, which means the store has to give crazy staple guy $30 back.
If you add together everything on the receipt, crazy staple guy owes $60.00. So he pays and you can finally buy your fucking milk, right? Nah.
"Wait," says crazy staple guy, "I can't afford that. Take 40 of the boxes off the receipt."
"OK," says the cashier. "When I first scanned the staples, I wrote 100 x (90 cents) on the receipt, so when I take off 40 boxes, I'll put -40 x (90 cents) on the receipt. That comes to -$36.00." It's a negative number because the store has to give this amount back to crazy staple guy.
The manager comes by and says, "Don't forget that he used all those coupons."
"Oh, right, duh," says the cashier. "I have to reverse the coupons too."
"When I first processed the coupons, I recorded 100 x (-30 cents) = -$30.00. So when I reverse 40 of the coupons, I'll record (-40) x (-30 cents), which is..."
Well, what is it? It's either $12.00 or -$12.00. But if it's -$12, that means the store has to give crazy staple guy $12. That makes no sense. When crazy staple guy used the coupons in the first place, he got money from the store. Now he's un-using the coupons, so he should be paying that money to the store. That means that -40 x (-30 cents) should be positive $12.00. It's positive because crazy staple guy has to give that much money to the store.
Adding together all the numbers, staple guy has to pay:
$90 from the original purchase -$30 from the original coupons -$36 from returning the staples $12 from getting the coupons back
The total is $36, which is the correct amount for someone buying 60 boxes of staples (after the return) at 60 cents each (after the coupon).
Note that if we decided that negative times negative should be negative, then we would have calculated:
$90 from the original purchase -$30 from the original coupons -$36 from returning the staples -$12 from getting the coupons back
for a total of $12. That's not what the charge should have been. In other words, getting this wrong means that the store gets ripped off, so yes, there is a real application.