# Multiplying halves by halves: Secrets of the Mathematical Ninja

The student, by now, knew better than to pick up the calculator. “There’s got to be a way of doing that quickly.”

The Mathematical Ninja looked a little disappointed; he’d built an elaborate electromagnetic pulse generator for the express purpose of killing calculators in a mysterious flash. “You’re right,” he said, grudgingly.

“$7.5 \times 3.5$,” he muttered again, as if asking for a clue.

“Brackets”, sotto-voced the Mathematical Ninja, helpfully.

“Oh… so do it as $(7 + \frac12)(3 + \frac12)$? That… give me a second.” He reached for a pen and paper. “That’s 21, plus half of seven, plus half of 3, plus half of a half. 21 + 3.5 + 1.5 + … is it 0.25?”

“We call it a quarter where I come from,” said the Mathematical Ninja. “Yep, 26 and a quarter. There’s a slightly quicker way, though - rather than work out a half of seven and a half of three separately, you can glom them together and say ‘half of ten’.”

“Nice… but doesn’t that only come out nicely if they’re both odd or both even?”

The Mathematical Ninja was, quietly, impressed; this was a slightly intelligent question.

“So, if you had $7.5 \times 4.5$, you’d need to work out half of 11, which is a pain.”

“It’s not *that* bad,” said the Mathematical Ninja. “But, if you didn’t like it, you could do it as $(8 - \frac12)(4 + \frac12)$, so you’d have 32, plus half the *difference*, *minus* a quarter.”

“35.75?” said the student, reaching for his calculator to check. “No, wait! 33.75!” ((Thanks, Claudio, for the correction.))

The Mathematical Ninja let it pass, just this once.