Does 0.999… really equal 1?

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This is the episode's narration, word for word. Headings jump to that point in the video.

Less than one, or one? 0:00

Here's a puzzle. Write zero, a decimal point, and then nines, forever. Zero point nine recurring. Is that equal to one, or just a little bit less than one? Most people's gut says less. Hold on to your answer, because by the end you'll know exactly what those three little dots mean.

The case for less 0:23

Your gut has a good argument. Zero point nine is a tenth short of one. Zero point nine nine is a hundredth short, and zero point nine nine nine is a thousandth short. However many nines you write down, you're always short by a little. So surely the recurring version is short by a little too? Keep that thought, because it's exactly where the trouble hides.

The Warwick questionnaire 0:50

You're in good company. In 1978, David Tall and Rolph Schwarzenberger published what they'd found when they asked new maths students at the University of Warwick this very question. Most said less than one. One wrote that it's "the nearest you can get to one without actually saying it is one". Another said they're the same, "because the difference between them is infinitely small". A third said just less than one, because "the difference between it and one is infinitely small". The last two used the same reasoning and came to opposite answers.

One third, times three 1:29

Here's the argument many people find the most convincing. Divide one by three and you get zero point three recurring. Three thirds make one, and three lots of zero point three recurring make zero point nine recurring. So they must be equal. It's convincing, but notice what it assumes: that a never-ending decimal equals a fraction. That's the very thing we doubted, so this is a check, not a proof. Other researchers report students who accept the threes but reject the nines.

Ten times, take away 2:04

The second argument is algebra. Call the number x. Ten times x is nine point nine recurring. Take x away, and the endless tails of nines cancel, leaving nine x equals nine, so x is one. One Warwick student asked the obvious question: when you multiply by ten, what happens to the nine at the right-hand end? There is no right-hand end. But the algebra quietly treats an endless decimal as an ordinary number you can multiply and subtract. Again, that's what we're trying to establish.

A book for wine-gaugers 2:43

So where did these endless decimals come from? In 1585, Simon Stevin, born in Bruges, published a little book, De Thiende, The Art of Tenths. It was written for surveyors, stargazers, carpet-makers, wine-gaugers and merchants. It's been called the first printed treatise on decimal fractions. But Stevin didn't invent decimals. The earliest known text to write them the way we do, digit by digit with a mark for the decimal point, is by al-Uqlidisi, written in Damascus around 952. Chinese mathematicians were using decimal fractions centuries earlier still, and Arab mathematicians long before Stevin.

Stevin meets one third 3:32

Stevin's notation looks odd to us: a little circled number after each digit to mark its place. And in the same little book, he hit our problem. Divide by three and the threes keep coming, he wrote, "infinitly". He wrote down the exact answer too, keeping the leftover third in the last place. But his advice was practical: you may come "so neere as the thing requireth", and stop. Nobody in business, he said, reckons in thousandths of a grain. But it left a question hanging: what does a decimal that never stops actually mean?

What the dots promise 4:13

Here's the real puzzle. Each place in a decimal is worth a tenth of the place before. So zero point nine recurring means nine tenths, plus nine hundredths, plus nine thousandths, and so on, with no last term. But you can't actually do infinitely many additions. Nobody can. So before we can ask whether that sum equals one, we need to say what an endless sum even is.

Cauchy's definition 4:42

The answer that stuck came from Paris, in 1821. Augustin-Louis Cauchy wrote a course for École Polytechnique students, as rigorous as possible. In it he defined a limit. When values approach a fixed value, so that in the end they differ from it by as little as you like, that fixed value is their limit. And the sum of an endless series, he said, is the limit of its running totals.

Running totals 5:13

So take the running totals of our nines: zero point nine, zero point nine nine, zero point nine nine nine. Each one falls short of one by a tenth, a hundredth, a thousandth, shrinking past every size you can name. Now Cauchy's definition does the work: the value those totals close in on is the limit, and the limit here is one. Mathematicians call a sum like this, where each term is the same fraction of the last, a geometric series, and the textbook formula, which is itself proved with limits, gives the same answer: exactly one. So the three dots don't mean keep writing nines; they're the name of that limit.

The schoolteacher 6:01

Cauchy's wording, values that "approach" a limit, still sounds like motion. The precise style that's taught today owes most to Karl Weierstrass, who spent fifteen years teaching in provincial Prussian schools before Berlin hired him in 1856. He's been called the father of modern analysis.

The challenge game 6:27

The modern definition works like a game. You name a distance, as tiny as you like. I have to show that, past some point, every running total is closer to the limit than your distance. Say within a millionth. Seven nines does it, because zero point nine nine nine nine nine nine nine is short by one ten-millionth. A billionth? Ten nines. Whatever you name, I win, so the limit is one, and by definition so is zero point nine recurring.

No room in between 7:03

Here's another way in. If two numbers are different, there's room between them, their average for instance. So try to name a number between zero point nine recurring and one. Suppose there were a gap, however small. Write enough nines and you're already closer to one than that gap, and the recurring number is bigger still, so the gap can't exist. Its size has to be zero. The rule doing the work is that any gap, however tiny, is bigger than a tenth, or a hundredth, or a thousandth… if you go far enough. Mathematicians call that the Archimedean property: in the ordinary numbers, nothing is infinitely close to one without being one.

What is a number, anyway? 7:48

But there was a crack under all of this. Cauchy said an irrational number is a limit of fractions, while assuming the numbers were already there. In the autumn of 1858, Richard Dedekind was teaching calculus in Zürich, and felt "the lack of a really scientific foundation for arithmetic". He later claimed nobody had ever properly proved that root two times root three is root six. He found his answer on the twenty-fourth of November, 1858, then left it unpublished until 1872. That same year, Georg Cantor published another way to build the numbers.

Building the line 8:33

Dedekind's idea: a real number is where you cut the fractions in two, the ones below and the ones above. Now cut at zero point nine recurring, and at one. Every fraction less than one is passed by some running total, so it lands below both cuts. The two cuts are identical, so the numbers are identical. In Cantor's version, numbers are built from sequences of fractions that close in. The sequence of running totals and the sequence one, one, one close in on each other, so again they're the same number. The equality isn't a trick. It's built into the foundations.

Why your gut still says no 9:18

If your gut still says no, the researchers have a list of why. Tall and Schwarzenberger saw students read the dots as a large but finite number of nines. They imagined a difference that's infinitely small but not zero. And in everyday speech, close means not equal, or we'd just say equal. A later study of a hundred and twenty university students, as other researchers report it, found many calling it the number next to one. At the University of Tasmania, Kim Beswick says that if you ask a class to put both numbers in order, controversy is virtually guaranteed.

A process, or a place? 10:00

The deepest trap is picturing zero point nine recurring as something moving, a dot creeping towards one and never arriving. That describes the running totals, not the number. Would you call one third a process, just because its decimal never ends? Mathematicians have built number systems that do contain infinitely small numbers, but they're different systems, and they give up the Archimedean property.

Two names, one number 10:29

And one isn't the only number with two names. Every decimal that stops has a twin that ends in endless nines. A half is zero point five, and also zero point four nine recurring. Two point three one seven is also two point three one six nine recurring. Decimals that never stop, like one third, have just the one form. It's like a half and two quarters: different spellings of the same number. That's a surprise that trips up many students: they expect every number to have exactly one decimal.

So, does it? 11:06

So, does zero point nine recurring really equal one? Yes, exactly. Not nearly, not rounded. The three dots are shorthand for a limit, and the limit is one. The puzzle was never really about numbers. It was about notation, and it took mathematicians until the eighteen hundreds to say precisely what that notation means. Next time, a puzzle where intuition fails just as hard. Three doors, one prize, and a host who knows where it is. Should you switch doors?

Sources

Every factual claim in the episode is tied to one of these. Spotted an error? Tell us.

1. D. O. Tall and R. L. E. Schwarzenberger, "Conflicts in the Learning of Real Numbers… — web.archive.org
2. Kim Beswick, "Why Does 0.999… = 1? A Perennial Question and Number Sense", Australian… — files.eric.ed.gov
3. Wikipedia, "Rolph Ludwig Edward Schwarzenberger" — en.wikipedia.org
4. Anderson Norton and Michael Baldwin, "Does 0.999… Really Equal 1?", The Mathematics… — files.eric.ed.gov
5. J. J. O'Connor and E. F. Robertson, "Simon Stevin", MacTutor History of Mathematics,… — mathshistory.st-andrews.ac.uk
6. Kathleen M. Clark, "In these numbers we use no fractions: A Classroom Module on… — web.archive.org
7. Simon Stevin, De Thiende, Leiden: Christoffel Plantijn (1585), title page and page 13,… — commons.wikimedia.org
8. J. J. O'Connor and E. F. Robertson, "Al-Uqlidisi (920 - 980)", MacTutor History of… — mathshistory.st-andrews.ac.uk
9. J. J. O'Connor and E. F. Robertson, "Chinese overview", MacTutor History of… — mathshistory.st-andrews.ac.uk
10. Simon Stevin, Disme: the art of tenths, or decimall arithmetike, English translation… — archive.org
11. Augustin-Louis Cauchy, Cours d'analyse de l'École royale polytechnique, 1re partie:… — archive.org
12. OpenStax, Calculus Volume 2, section 5.2 "Infinite Series" — openstax.org
13. J. J. O'Connor and E. F. Robertson, "Augustin-Louis Cauchy", MacTutor History of… — mathshistory.st-andrews.ac.uk
14. J. J. O'Connor and E. F. Robertson, "Karl Weierstrass", MacTutor History of… — mathshistory.st-andrews.ac.uk
15. J. J. O'Connor and E. F. Robertson, "Real numbers 2" (The real numbers: Stevin to… — mathshistory.st-andrews.ac.uk
16. Richard Dedekind, "Continuity and Irrational Numbers" (1872), in Essays on the Theory… — gutenberg.org
17. J. J. O'Connor and E. F. Robertson, "Richard Dedekind", MacTutor History of… — mathshistory.st-andrews.ac.uk

Image credits

• Simon Stevin, De Thiende, Leiden: Plantijn, 1585, title page · public domain · via Wikimedia Commons
• Simon Stevin, De Thiende (1585), p. 13: adding decimals · public domain · via Wikimedia Commons

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