The Mathematical Ninja and The Slinky Coincidence

"No, no, wait!" said the student. "Look!"

"8.000 000 072 9," said the Mathematical Ninja. "Isn't that $\frac{987,654,321}{123,456,789}$? What do you think this is, some sort of a game?"

"It has all the hallmarks of..."

"I'll hallmark you in a minute!" said the Mathematical Ninja.

Seconds later, the students arms were above his head and a set of skittles had appeared from nowhere.

"Er, sensei? Ten out of ten for the Bond-esque bon mot, and everything, but I think you might lose some marks for it being a complete non-sequitur as far as the cartoon punishment goes. How about you focus on explaining the number trick?"

The Mathematical Ninja raised his eyebrows. "Sh," he said. "Playing." He swung the student at the skittles. "Leave your legs down! For heaven's sake."

"I can see it's a bit short of a billion divided by a bit short of 125 million," said the student, "so somewhere about 8 looks about right. But why so close?"

"It's to do with the binomial expansion," said the Mathematical Ninja, swinging again. "Feet still!"

"But those skittles are heavy and I've got dancing later. What about the binomial expansion?"

"Well, $(1-x)^{-2} = 1 + x + 2x^2 + 3x^2 + ...$. With $x = 0.1$, that gives $\frac{100}{81} = 1.\dot{1}234567\dot{9}$."

"I can see that," said the student, stretching to knock down a skittle on his own terms.

"Meanwhile, $\frac{100}{9} = 11.\dot{1}$."

"As any fule no."

"And $\frac{100}{9} - \frac{100}{81} = \frac{800}{81} = 9.\dot{8}7654320\dot{9}$."

"I'd need to write that down, sensei, but I'll take your word for it."

"The upshot is that $\frac{9.87654320}{1.23456790}$ is eight, and the number you gave, $\frac{987654321}{123456789}$ is a tiny fraction larger on top and a tiny fraction smaller on the bottom."

"So it's not just a coincidence, it's based on something solid?"

"STRIKE!" screamed the Mathematical Ninja. "There are no coincidences."

* Edited 2016-04-04 to correct a typo. Thanks, @christianp!

Colin

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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