When .999... Isn't 1

In ordinary math, the infinite decimal .999… is defined to be the limit of the terminating decimals .9, .99, .999, …; that is, it’s defined to be the real number that the fractions 9/10, 99/100, 999/1000, … approach in the ordinary sense. And that limit is most definitely 1, not some real number that’s a tiny bit less than 1. This is not an approximate truth; it’s a 100% accurate, rigorously established mathematical fact. It’s a part of how the real number system works, and it’s a feature, not a bug.

But if you’ve read my essay “Marvelous Arithmetics of Distance”, you already know that there are number systems in which things that look like rational numbers behave differently¹, and you won’t be too surprised to learn that there’s another new number-game to play. It’s the q-deformed real number “game” of Sophie Morier-Genoud and Valentin Ovsienko, and in the context of their work, it becomes true (in an admittedly somewhat arcane sense) that 9/10, 99/100, 999/1000, etc. do not approach 1 but rather approach something smaller. Except that it’s not the numbers themselves that behave in this ill-bred way; it’s their avatars in the q-deformed world, avatars that Morier-Genoud and Ovsienko write as [9/10]_q , [99/100]_q , [999/1000]_q , etc.

In this essay, without going too deeply into the underlying theory, we’ll take a concrete look at the q-deformations of the numbers 1/2, 2/3, 3/4, etc. and of the numbers 2/1, 3/2, 4/3, etc. In the ordinary real number system, all these fractions approach 1 as the numerators and denominators get big, regardless of whether the bigger number is on top of the fraction or on the bottom. But looking at these rational numbers through the spectacles that Morier-Genoud and Ovsienko have given us, we’ll see that we get two different limits according to whether the numerator is bigger than the denominator or vice versa. In particular, we’ll find that, while the q-deformations [2/1]_q, [3/2]_q, [4/3]_q, etc. approach [1]_q, the q-deformations [1/2]_q, [2/3]_q, [3/4]_q, etc. approach [1]_q’s evil (or maybe not so evil) twin.

Morier-Genoud and Ovsienko’s work continues a centuries-old tradition of taking facts about numbers and upgrading them to facts about functions, often called generating functions in this context. Traditionally, the independent variable in these functions is called q. For instance, a common q-analogue of the number n! (defined as the number 1×2×3×···×n) is the polynomial function (1)(1+q)(1+q+q²)···(1+q+q² +···+qⁿ⁻¹), the “q-factorial” of n, which becomes n! when you replace q by 1.² Some ways of sticking a q into a formula work; others don’t. Seeking the right q-analogue of some known bit of math can be an adventure. Certain mathematicians, seeing their colleagues succumb to the temptation of adding an extra variable or two, jestingly refer to “the q disease”. And now I’m infecting you with it.³

If this were a more historical essay I’d talk about where q comes from, as a confluence of eighteenth century combinatorics, nineteenth century number theory, and twentieth century physics. But I’m assuming you’re more interested in math than math history, so let me just assure you that the q you’re about to meet has an illustrious pedigree.

RAMBLING THROUGH THE RATIONALS

There are several ways to define the q-deformed rational and q-deformed real numbers, none of them easy (which is probably why they weren’t discovered till recently). But it’s possible to play with them without defining them conceptually, by giving rules for computing them. Each of our new “numbers” is going to be a finite or infinite expression involving a variable q; we call q a “formal indeterminate” as a way of signaling that you’re not supposed to regard q as having any particular value. For instance, [1/3]_q = q²/(1+q+q²). The kind of function of q that we see when we look at the q-deformed rational numbers is what’s called a rational function; it’s a function of q that can be computed using only addition, subtraction, multiplication, and division. The definitions of [n]_q when n is an integer is centuries old, but the correct way to define [r]_q when r is not an integer only came to light in the past decade, through the work of Morier-Genoud and Ovsienko.

Here are three rules that taken together and patiently applied will allow you to compute [r]_q for any rational number r:

(1) [0]_q = 0

(2) [r+1]_q = q[r]_q + 1

(3) [r]_q [−1/r]_q = −1/q

For instance, applying (2) with r = 0 and using (1) you can calculate

[1]_q = 1

Now applying (2) with r = 1 you can calculate

[2]_q = 1+q

Now applying (3) with r = 2 you can calculate

[−1/2]_q = −1/(q+q²)

Next applying (2) with r = −1/2 you can calculate

[1/2]_q = q/(1+q)

And so on, round and round. You can ramble through the rational numbers for a long time, and you may encounter some rational numbers you’ve seen before, but the rules will never lead to a contradiction. For instance, by applying rules (2) and (3) in alternation, you can go from 1/2 to 2/3 to −2/3 to 1/3 to −3 to −2 to 1/2, and you’ll find that our rules bring us back to [1/2]_q = q/(1+q). Here’s a picture of what sorts of steps we can take in the rational numbers, where I’ve drawn the rational numbers on a “number circle” rather than on the usual number line, with 0 at the bottom. Blue edges correspond to rule (2) and red edges correspond to rule (3).

Here’s a roadmap for rambling through the rationals.
There should be infinitely many blue and red lines,
but then all you’d see would be a red oval in a purple circle.

The surprising consistency of these rules when we go around a loop is a clue that something important is going on.

Take a look at what we’ve learned so far:

[1]_q = 1

[2]_q = 1+q

[1/2]_q = q/(1+q)

[−1/2]_q = −1/(q+q²)

Notice that if you replace q by 1, [r]_q simplifies to r. This is what we mean when we say that [r]_q is a “deformation” of r; when q is 1, [r]_q is just good-old r.

BOTH SIDES NOW

With rules (1), (2), and (3), you can show that

[1/2]_q = q/(1+q)
[2/3]_q = (q+q²)/(1+q+q²)
[3/4]_q = (q+q²+q³)/(1+q+q²+q³)
…

and that

[2/1]_q = (1+q)/1
[3/2]_q = (1+q+q²)/(1+q)
[4/3]_q = (1+q+q²+q³)/(1+q+q²)
…

Let’s look at the improper fractions first. Multiplying numerators and denominators by 1−q, we can rewrite [2/1]_q as (1−q²)/(1−q), [3/2]_q as (1−q³)/(1−q²), [4/3]_q as (1−q⁴)/(1−q³), and more generally rewrite [(n+1)/n]_q as (1−qⁿ⁺¹)/(1 − qⁿ). What can we say about how [(n+1)/n]_q behaves when n gets large, assuming that q is some fixed number strictly between −1 and 1? In this case qⁿ and qⁿ⁺¹ both approach zero, so the fraction (1−qⁿ⁺¹)/(1−qⁿ) approaches (1−0)/(1−0), or 1. And remember, [1]_q is also 1. So [2/1]_q, [3/2]_q, [4/3]_q, . . . converge to [1]_q , as we would hope. Pictorially:

The limit approached by the q-avatars of 2/1, 3/2, 4/3, etc.

It’s the proper fractions that don’t behave properly when −1 < q < 1. The source of the impropriety is the way that the sums in the numerators of [n/(n+1)]_q start with “q+q²+…” instead of “1+q+…”. Multiplying numerators and denominators by 1−q, we can rewrite [1/2]_q as (q−q²)/(1−q²), [2/3]_q as (q−q³)/(1−q³), [3/4]_q as (q−q⁴)/(1−q⁴), and more generally rewrite [n/(n+1)]_q as (q−qⁿ⁺¹)/(1−qⁿ⁺¹). As n gets large, the fractions (q−qⁿ⁺¹)/(1−qⁿ⁺¹) approach (q−0)/(1−0), or q. So the q-deformed fractions [1/2]_q, [2/3]_q, [3/4]_q, … don’t converge to [1]_q, but rather to a different function of q, namely, q itself. Pictorially:

The limit approached by the q-avatars of 1/2, 2/3, 3/4, etc.

Likewise you can show that [1/2]_q , [1/3]_q , [1/4]_q , . . . approach [0]_q but that [−1/2]_q , [−1/3]_q , [−1/4]_q , . . . approach a different function of q, namely 1−1/q (though we have to be careful not to let q equal 0).

Morier-Genoud and Ovsienko show that as long as the rational numbers r₁, r₂, r₃, … approach some rational limit s from above (or, in number-line terms, from the right), their q-deformations will behave well: the functions [r₁]_q, [r₂]_q, [r₃]_q, … will approach the function [s]_q, at least when q is close enough to 0. But if r₁, r₂, r₃, … approach the rational limit s from below (or, in number-line terms, from the left), their q-deformations [r₁]_q, [r₂]_q, [r₃]_q, … will not approach the function [s]_q but will instead approach a different deformation of s – a function of q that is actually smaller than [s]_q when q is near 0, even though the two functions agree at q = 1. Researchers Asilata Bapat, Louis Becker, and Anthony Licata represent this “twin” of [s]_q by the symbol [s]^♭_q, and write the ordinary [s]_q as [s]^♯_q when they want to stress the parallelism.

FILLING THE HOLES

So far I’ve only talked about the q-deformed rational numbers, but what about q-deformed irrational numbers, filling the holes in our strange new number line? What happens when the rational numbers r₁, r₂, r₃, … approach an irrational limit s? In that case, it turns out not to matter whether the r_n’s approach s from the left, from the right, by oscillation, quickly, slowly, whatever – in every case, the q-deformed rationals [r₁]_q, [r₂]_q, [r₃]_q, … will converge to one specific function of q, which Morier-Genoud and Ovsienko call the q-deformation of the irrational number s, written as [s]_q.

For example, just as the fractions 2/1, 3/2, 5/3, 8/5, . . . (ratios of consecutive Fibonacci numbers) converge to the celebrated golden ratio

Φ=(1+sqrt(5))/2,

the q-deformed fractions [2/1]_q, [3/2]_q, [5/3]_q, [8/5]_q, … converge to the function

[Φ]_q = (q²+q−1+sqrt(q⁴+2q³−q²+2q+1))/(2q)

For more of a woo-woo vibe, you can call this “quantum Φ” instead of “q-deformed Φ”.

Want to learn about quantum sqrt(2) and quantum e and quantum π? Check out what Morier-Genoud and Ovsienko have written. Or you can learn about the q-deformed real numbers by watching online talks the two have given courtesy of the One World Numeration Seminar.

DEJA VU?

The division of the q-rational numbers into sharps and flats is curiously reminiscent of many students’ mistaken beliefs about infinite decimals, and in particular the belief that even though the sequence 1.1, 1.01, 1.001, … converges to 1, the sequence 0.9, 0.99, 0.999, … does not converge to 1 but rather converges to 0.999… conceived of as a “smaller 1”. It may also remind some of you of the literal decimal system I described in my essay “More About .999…”); in that system, 0.999… and 1.000… are genuinely different. So you might be tempted to relate the q-deformed rationals to the literal decimals. But there’s an important mismatch between the q-deformed integers and the literal decimal system. In the literal decimal system, some rational numbers (like 1/2 and 1/5) have an evil twin but others (like 1/3) don’t; different rational numbers are treated differently according to how their denominators factor into primes, with 2 and 5 (the prime factors of 10) being accorded special treatment. In contrast, the society of q-deformed rational numbers is an egalitarian one: every rational number, regardless of numerator or denominator, has two avatars, one ever so slightly smaller than the other.

A more important difference is that I brought the literal decimals out of obscurity and into the blogosphere not because they’re useful (they’re not) or beautiful (I don’t think so) but because I wanted to make a point about the freedom of mathematics: you can change the rules as long as your new rules are consistent and you’re honest about having changed the rules. But just because you’re allowed to define a new number system doesn’t mean that you should, or that if you do, that anybody else should care! In comparison, the q-deformed rational number system is “real mathematics”, with connections to topics like continued fractions, quantum groups, and knot theory. The q-rationals wanted to be found, and one of the more amazing things about them is that nobody found them sooner.

If you’re looking for precedents for the strange behavior of the sharps and flats, a better analogy can be made with Dedekind cuts, as described in my essay “Dedekind’s Subtle Knife”. When we construct the real numbers via Dedekind cuts of rational numbers, we get a single cut corresponding to each irrational number, but if we’re not careful we’ll wind up with two copies of each rational number r, corresponding to the two ways to cut the rational number-line at r (with r itself going into either the left set or the right set). There are ways to fix this when one constructs the reals via Dedekind cuts, but in the case of the q-deformed rational numbers, we don’t view the duplication of the rationals as a bug to be fixed; we accept it as a fact of life in our strange new number system, just as non-duplication (e.g., .999… = 1) is a fact of life in the ordinary reals.⁴

FURTHER WEIRDNESS

One fascinating feature of the q-deformed real numbers is that, when they’re expressed as power series around 0 in the variable q, all the coefficients are integers; for instance, the q-deformed golden ratio [Φ]_q expands as

1+q²−q³+2q⁴−4q⁵+8q⁶−17q⁷+37q⁸−82q⁹+185q¹⁰−423q¹¹+978q¹²−…

You may not recognize the sequence of coefficients, but the Online Encyclopedia of Integer Sequences does, and http://oeis.org will gladly tell you that this is a sequence of “generalized Catalan numbers”, which count something. So the golden ratio – not a rational number, let alone a counting number! – has hidden combinatorial meaning.

Even though the power series for [Φ]_q diverges when you plug in q = 1, evaluating the algebraic expression for [Φ]_q at q = 1 still gives Φ and still has a sort of combinatorial meaning, best expressed in probabilistic language: it says that if you take a half-infinite strip of height 2 that starts at the left and goes infinitely far to the right and you tile it by dominos “at random” (whatever that means!) with no gaps or overlaps, then the odds are Φ-to-1 that the two leftmost squares are covered by a single vertical domino (as in the picture) rather than two horizontal dominos.

A random domino tiling of a half-infinite strip (whatever that means!).

THE TWISTY ROAD AHEAD

The study of q-deformed rational numbers and q-deformed real numbers is still in its infancy. One thorny challenge is bringing ordinary addition and ordinary multiplication fully into the q-deformed world, by which I mean devising operations +_q and ×_q on the power-series representations of the q-functions so that [r]_q +_q [s]_q = [r+s]_q and [r]_q ×_q [s]_q = [rs]_q . For instance, since [Φ]_q is 1+q²−q³+2q⁴−…, you might think that [2Φ]_q should be 2+2q²−2q³+4q⁴−…, or perhaps be 1+q times 1+q²−q³+2q⁴−…, but no: the power series expansion of [2Φ]_q begins 1+q+q²+q⁷−q⁹−….

It’ll probably be years before we can say we understand what the q-deformed real numbers are trying to tell us, and possibly decades before we can say what they’re good for. But I want to spread the word about this work now because the q-deformed reals offer a kind of partial vindication to anyone who ever learned to live with “.999… = 1” but never learned to like it, or who learned to like it but still has fond memories of .999… from the days when it seemed to be its own numinous thing rather than just 1 trying to look mysterious. Of course every mathematician must at some point in their education come to grips with the topology of the real line and take to heart the limitations of infinite decimals as stand-ins for the real numbers themselves. But I think it’s fun that there’s a respectable kind of math in which the sequence [1/2]_q, [2/3]_q, [3/4]_q, … and the sequence [2/1]_q, [3/2]_q, [4/3]_q, … converge to different limits.

It just goes to show that the mathematical universe is weirder than you think it is, even when you know that it’s weirder than you think it is.

Thanks to Sophie Morier-Genoud, Valentin Ovsienko, Sandi Gubin, and Nick Ovenhouse.

ENDNOTES

#1. For instance, the infinite sum 1+2+4+8+… diverges in the real numbers but converges 2-adically to −1.

#2. If you take the formula (a+b)!/(a!b!) for binomial coefficients and replace all three factorials by q-factorials, you get “q-binomial coefficients”, which play a role in the “q-binomial theorem”. And so on.

#3. As R. A. Lafferty says to the reader near the start of his novel “Fourth Mansions”: “You who glanced in here for but a moment, you are already snake-bit! It is too late for you to withdraw. The damage is done to you. … Die a little. There is reason for it.”

#4. The sharps and flats together have nice convergence behavior that mimics the way the two kinds of rational cuts behave under the set-theory operations of union and intersection. When a decreasing sequence of rational numbers r_n converges to some rational number s, then the [r_n]^♯_q’s converge to [s]^♯_q and the [r_n]^♭_q’s also converge to [s]^♯_q. On the other hand, when an increasing sequence of rational numbers r_n converges to some rational number s, then the [r_n]^♯_q’s converge to [s]^♭_q and the [r_n]^♭_q’s also converge to [s]^♭_q.

REFERENCES

Asilata Bapat, Louis Becker, and Anthony M. Licata, q-deformed rational numbers and the 2-Calabi–Yau category of type A₂, https://arxiv.org/abs/2202.07613.

Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, Experimental Mathematics (2019); https://arxiv.org/pdf/1908.04365.

Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and irrationals, preprint (2025); https://arxiv.org/abs/2503.23834.