The Mathematical Ninja's Rules of Fractions

The student, wisely, stammered an apology and the Mathematical Ninja pulled him back in through the window by the ankle. “But I just…”

“Shht.”

“I mean…”

“Shht.”

The Mathematical Ninja took a step towards the student and the student, finally shhted ¹.

“Never,” said the Mathematical Ninja, “let me catch you doing that to a fraction again.”

The student meekly nodded and looked again at $\frac{64(x^2+2)}{3(x+6)}$. He picked his words with the utmost care. “So, just to be absolutely clear, you never want to catch me - incorrectly,” he hastened to add, “trying to cancel the $x^2$ with the $x$ or the $2$ with the $6$?”

Eyes narrow, the Mathematical Ninja nodded. “Why not?”

“Because…” Think, little student, think! “Because… because…” (Don’t say ‘because of the wonderful things he does!) “Because you can only cancel FACTORS!” Phew!

The Mathematical Ninja’s eyebrows rose by a fraction, the closest he ever came to expressing impressedness. “The Mathematical Ninja’s Second Rule of Fractions,” he said, “is that you can only cancel factors. Full factors.”

“What’s the first rule of fractions?” Oldest trick in the book, student. Oldest trick in the book. You want someone to learn two rules? Teach them the second rule first.

“The Mathematical Ninja’s First Rule of Fractions,” said the Mathematical Ninja, “is that you may split up the top of a fraction, but never the bottom.”

“What about partial fractions?” asked the student, before remembering that not even the Mathematical Ninja likes a smartarse.

Footnotes:

1. He’d have been within his rights to shht himself.