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."


"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.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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