The Fraction Hiding in 142857

Type 1 ÷ 7 into a calculator. The screen fills with 0.14285714285714, the same six digits cycling for as long as the screen lets you read.

Those digits, in that order, appeared in an earlier article in this series. 142857 was presented as a curiosity that rearranges itself when multiplied. By 2, you get 285714. By 3, 428571. By 4, 571428. Same six digits, different order, every time, all the way to six. Multiply by 7 and they collapse into six nines.

It looked like maths doing a card trick. It was one-seventh.

The wheel

One-seventh, written as a decimal, is 0.142857142857142857, the same six digits cycling forever.

Multiply 1/7 by 2 and you are working out 2/7, which is 0.285714285714. The same six digits, just starting in a different place. 3/7 is 0.428571428571. Same digits, different start. So is 4/7. So is 5/7. So is 6/7.

The numbers were not shuffling. They were a wheel. Multiplying by 2, 3, 4, 5 or 6 just chose a different place on the wheel to start reading. Multiply by 7 and the fraction becomes 7/7, which is 1, and the wheel disappears. The six nines are what the decimal looks like when it tries to write the number 1 out of sevenths. The card trick is one fraction, read seven ways.

Why some fractions stop and others don't

Every fraction has a decimal twin. Some twins stop. Others run on forever.

One-half is 0.5. One-quarter is 0.25. One-fifth is 0.2. One-eighth is 0.125. These end and stay ended.

One-third is 0.333, threes going forever. One-sixth is 0.166, sixes going forever. One-seventh you have just met. One-ninth is 0.111, ones going forever. These never end.

The rule sits in the bottom of the fraction. Halves, quarters, fifths, eighths and tenths divide neatly into ten or its multiples, so their decimals finish. Thirds, sixths, sevenths and ninths do not, so theirs cannot.

We chose ten because we have ten fingers. The fractions did not vote.

One number, two scripts

Fractions and decimals are not two topics in maths. They are the same numbers written two ways. One-half and 0.5 are not equivalent the way "two" in English and "deux" in French are equivalent. They are the same word, written twice.

A child who understands that 1/7 and 0.142857142857 are literally the same number has done some of the real work of learning what a number is. Most don't.

It also explains the fact that 0.999 recurring equals 1, the curiosity from earlier in the series that has started more arguments than almost anything in maths. One-third as a decimal is 0.333 forever. Three times one-third is one. So three times 0.333 forever is 0.999 forever. The arithmetic was always going to land there.

The card trick was a fraction. The recurring decimals that look like glitches are just fractions whose denominators do not get on with ten. Maths has more synonyms than English does. It just doesn't tell you which words mean the same thing.