# Recurring decimals: Secrets of the Mathematical Ninja

“May I borrow some paper?” asked the student, meekly. He knew he should have come prepared; he feared for his safety, as the Mathematical Ninja’s reputation preceded him.

“BORROW?!” hairdryered the Mathematical Ninja. “BORROW?! Why on earth would I want the paper back after you’ve defiled it with your profane… scratchings?”

“I’ll take that as a no,” muttered the student.

“What do you want it for?”

“I have to convert $0.\dot 3 \dot 6$ into a fraction.”

The Mathematical Ninja paused for a moment. “Since you’re going the correct way, I shall allow you to have some paper.”

“The correct way?”

“Yes. From ugly decimals to lovely fractions.”

“But I thought you always estimated things using…”

“SILENCE!” said the Mathematical Ninja.

“Normally, I’d say that it’s got two decimal places, so I’d multiply by 100 - but that just gives me $\dot 3 \dot 6$.”

The Mathematical Ninja cleared his throat. “Does it?”

“Evidently not. Well… $0.\dot 3\dot 6$ is the same as $0.3636363636…$”

“(You can stop there).”

“So, if I multiply it by 100, I get $36.363636…$”.

“Good,” begrudged the Mathematical Ninja. “So $x = 0.\dot 3\dot 6$ and $100x = 36.\dot 3 \dot 6$.”

The student diligently wrote it down on his recently-acquired paper. “And can I… take those away?”

“Do it.”

“$99x = 36$. Oh! That’s just algebra.”

“*Just*?”

“Erm… it’s my favourite thing!” he lied. “Divide both sides by 99? $x = \frac{36}{99}$.”

“Does it simplify?”

“Obviously, or else you wouldn’t ask. Cancel the 9s?”

“*Divide top and bottom by* 9,” said the Mathematical Ninja, thinking he’d been unwise to lend out so much of his weaponry.

”$\frac {4}{11}$”, said the student, triumphantly.