Thirteenths (Part 2/3): Secrets of the Mathematical Ninja

This is the second of a three-part series about working out thirteenths.

In the first part, you learned that the first step of finding thirteenths was to multiply by 77. The second part is to work out the nine’s complement of one less than your number.

What’s the nine’s complement?

Slight diversion: When I broke the world record that never was in 2012, I made extensive use of nine’s complements. All you do is figure out what digit you need to make a number up to 9: so the nine’s complement of 6 is 3, the nine’s complement of 8 is 1, and the nine’s complement of 0 is 9. And vice versa, in all cases.

Last time, trying to find 7/13, we’d worked out that $7 \times 77 = 539$, so we drop the number by 1 (538) and find the nine’s complement of each digit in turn: 461. Write down the two numbers next to each other and put a dot at either end; you’re done!

$\frac{7}{13} = 0.\dot{5}3846\dot{1}$

With a little practice, you’ll find you can do them in your sleep. Next time, I'll show you why they work.

* Edited 2017-04-07 to add links


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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