The sun is setting outside the dojo. The Mathematical Ninja's student knows this, for he is looking out of the window.
At some point, he notices the compass point a fraction of a centimetre from his eyeball. "Pay attention," says the Mathematical Ninja. "What does this add up to?"
He was meant to be paying attention to a factor theorem question with a bundle of negative signs knocking about. It was a bit blurry, since there was a compass in his immediate field of vision; he also took this as a signal that reaching for the calculator would be a poor tactic.
He had $f(x) = 6x^3 + 13x^2 - 79x - 140$ and wanted the remainder when divided by $(x+4)$. He knew, of course, to work out $f(-4)$ - he wasn't a complete idiot - but had drifted off in working out $6 \times - 64 + 13 \times 16 - 79 \times -4 - 140$.
He'd got as far as $-384 + 208 + 316 - 140$ without too many problems but those minus signs had got his goat. He explained this to the Mathematical Ninja, who scoffed, and put down the compass.
"It's easy," he said, as he always said. "You're probably trying to do crazy stuff like $-384 + 208$, right?"1 .
The student nodded.
"Don't do that. Just take the plus numbers and add them up…"
"$208 + 316 = 524$?"
He wasn't sure whether the "Very good" was sarcastic or genuine. "Now do the same with the negative numbers - just ignore the minus signs."
"$384 + 140 = 524$ as well! So I take the negative total from the positive total to get the sum?"
"And that always works?"
"As long as you've turned it into an add and subtract sum, absolutely."
"Thanks, Mathematical Ninja!"
* No eyeballs were harmed in the writing of this article.