A series that converges to minus one

The square that half, a quarter and an eighth fill ended on a provocation. The formula $1 + r + r^2 + \cdots = 1/(1 - r)$ holds for $|r| < 1$, and at $r = 2$ it returns $1/(1 - 2) = -1$: the terms $1, 2, 4, 8, \dots$ “ought” to add to minus one. That essay noted that the claim becomes true in a different number system and left it there. This essay follows it.

The claim is not a trick of formal manipulation. It is a statement about convergence, correct in a precise sense, and the sense comes from changing one thing: what it means for a number to be small. The geometric series has only ever needed one property of its ratio — that its powers shrink to nothing — and there is a way of measuring size under which the powers of 2 do.

Small means divisible by a high power of two #

Define the 2-adic size of a whole number as follows: write it as $2^k$ times an odd number, and its size is $2^{-k}$. So every odd number has size 1, the even numbers not divisible by 4 have size $\tfrac12$, multiples of 4 but not 8 have size $\tfrac14$, and so on. Zero has size 0. The size extends to fractions — $2^k$ times a ratio of odd numbers has size $2^{-k}$ — and it multiplies: the size of a product is the product of the sizes.

Under this measure $2^n$ has size $2^{-n}$, which goes to zero. So the powers of 2, which grow without bound in the ordinary sense, shrink to nothing 2-adically. That is exactly what the geometric series needs.

The partial sums are $1, 3, 7, 15, 31, \dots$, and the $n$-th is $2^n - 1$. Its distance from $-1$ is the size of $2^n - 1 - (-1) = 2^n$, which is $2^{-n}$. So the partial sums close in on $-1$, halving the distance at every step — the 2-adic mirror image of the square, where the uncut corner halved at every step. The figure checks the exact power of 2 in each difference with whole-number arithmetic, for twenty terms.

The same thing happens with any prime. Measuring size by powers of 3, the partial sums of $1 + 3 + 9 + 27 + \cdots$ are $(3^n - 1)/2$, and their distance from $-\tfrac12$ is the size of $3^n/2$, which is $3^{-n}$. The series converges 3-adically to $1/(1 - 3) = -\tfrac12$, exactly as the formula says.

Why the formula must hold #

Nothing new has to be proved to make the claim respectable. The identity

$$(1 - r)(1 + r + r^2 + \cdots + r^n) = 1 - r^{n+1}$$

is algebra, true for any number in any number system. Divide by $1 - r$, and the partial sum is $1/(1 - r)$ minus $r^{n+1}/(1 - r)$. The partial sums converge to $1/(1 - r)$ exactly when that last term goes to zero — exactly when the powers of $r$ go to zero. The comparison test and every other refinement of the real theory are built on top of this one step, and the step does not care what “goes to zero” means, provided the notion of size is compatible with multiplication. The 2-adic size is: the size of a product is the product of the sizes, so the size of $2^{n+1}$ is $2^{-(n+1)}$, and it goes to zero.

There is also a direct check that $-1$ is the right answer. Add 1 to the infinite binary string $\dots 1111$: the last digit becomes 0 with a carry, the next becomes 0 with a carry, and so on for ever, leaving $\dots 0000$. So $\dots 1111$ is the number which, increased by 1, gives 0 — which is what $-1$ means.

A strange geometry of closeness #

The 2-adic distance between two whole numbers is the 2-adic size of their difference. Drawn for the numbers 0 to 15, it looks nothing like ordinary distance.

Neighbours are far apart: consecutive numbers differ by 1, an odd number, so they are at distance 1, the maximum. Ordinary closeness is about agreeing in the leading digits; 2-adic closeness is about agreeing in the final ones, and the grid is what that looks like. Numbers that differ by 8 are close. And the distance satisfies a much stronger rule than the ordinary triangle inequality: the distance from $a$ to $c$ is at most the larger of the distances from $a$ to $b$ and from $b$ to $c$, not their sum. That is the strong triangle inequality, and a consequence of it — checked in the figure on every triple — is that every triangle is isosceles, with its two longest sides equal.

The strong inequality makes convergence easy in a way it never is for real numbers. A series converges in this distance as soon as its terms shrink to zero, whatever their signs and however slowly. The harmonic series, which needs a careful argument to diverge in the ordinary sense, has no analogue here: there is no “slowly shrinking terms that still add to infinity”. The subtleties of real series — conditional convergence, rearrangement, boundary cases — all disappear.

The 2-adic numbers as a tree #

Whole numbers can be arranged by their 2-adic closeness in a tree.

Read a number’s binary digits from the right, and use them as directions: at the first level go left for a 0 and right for a 1, at the second level do the same with the next digit, and so on. Two numbers share a path for exactly as many levels as they have final binary digits in common — that is, as the power of 2 dividing their difference — so the tree is a picture of the 2-adic distance. The leaves of a tree of depth $k$ are the remainders modulo $2^k$, the numbers on a clock with $2^k$ hours.

The 2-adic integers are the infinite paths down this tree: infinite strings of binary digits extending to the left. Every ordinary whole number is one of them, with zeros from some point on. But there are more, and $-1$ is the first example: in 2-adic arithmetic, $-1 = \dots 1111$, because adding 1 to it carries forever and leaves $\dots 0000 = 0$. The partial sums $1, 11, 111, 1111$ in binary are the paths $1, 3, 7, 15$, climbing down the rightmost branch towards it.

Minus one inside every computer #

The digits $\dots 1111$ for $-1$ are not only a 2-adic curiosity; they are how nearly every computer stores negative numbers. A machine integer of 8 bits holds a remainder modulo $2^8 = 256$, and in the convention called two’s complement the number $-1$ is stored as $11111111$ — the last eight digits of the 2-adic $-1$. The number $-3$ is stored as $11111101$, the last eight digits of the 2-adic expansion of $-3$, and adding $-1$ and $-3$ by the ordinary rules of binary addition, discarding the carry out of the top, gives $11111100$, which is $-4$. A 32-bit or 64-bit integer is the same thing with more digits: the last 32 or 64 digits of a 2-adic number.

That is why two’s complement arithmetic works without any special rules for negative numbers. It is arithmetic in a clock with $2^{64}$ hours, and on that clock the number that 1 + 2 + 4 + … + $2^{63}$ equals, $2^{64} - 1$, is the same position as $-1$. The machine is computing with truncated 2-adic integers, and the partial sums of the series in the hero figure are the machine’s representations of $-1$ at increasing word sizes.

Repeating digits are geometric series, in both directions #

The 2-adic view gives a new reading of an old fact. Every repeating decimal is a fraction, and the reason is a geometric series:

$$0.\overline{142857} = 142857 \cdot (10^{-6} + 10^{-12} + \cdots) = \frac{142857}{10^6 - 1} = \frac{1}{7}.$$

The block of digits repeats to the right, each copy $10^{-6}$ times the last, and the series converges because $10^{-6}$ is small in the ordinary sense.

2-adically, fractions with odd denominators have expansions that repeat to the left. Take $\tfrac13$: its 2-adic digits are $\dots 10101011$, a final $11$ and then the block $10$ repeated forever to the left. That string is $3 + 8 \cdot (1 + 4 + 16 + \cdots)$ in the 2-adic sense — the block $10$, which is 2, placed at every other position, contributing $2 \cdot 4 \cdot (1 + 4 + 4^2 + \cdots)$. The ratio is 4, small 2-adically, and the series sums to $2 \cdot 4/(1 - 4) = -\tfrac83$; adding the final two digits, 11 in binary or 3 in ordinary notation, gives $\tfrac13$. The figure checks each expansion the other way, by multiplying twenty-four digits of it by the denominator and confirming that the product agrees with the numerator in all twenty-four places.

Why repetition happens at all is a pigeonhole argument in both directions. Long division by 7 can leave only seven different remainders, so some remainder recurs and the digits cycle; computing the 2-adic digits of $\tfrac17$ likewise passes through a bounded set of intermediate numerators, and the digits cycle again.

So the two facts are one fact. A string of digits that eventually repeats is a rational number in any base and in either direction, because a repeating block is a geometric series and a geometric series with a small ratio sums to a fraction. What changes between the real and the 2-adic readings is only which direction of repetition counts as small.

Where the series converges #

The geometric series with ratio $r$ converges wherever $r$ is small, and “small” now has several meanings.

A ratio $a/b$ in lowest terms is 2-adically small exactly when its numerator is even — then its denominator is odd and the ratio has at least one factor of 2. It is 3-adically small exactly when its numerator is a multiple of 3. These conditions have nothing to do with the ordinary one, $|a| < b$, and the figure’s three regions cut across each other. A ratio like 2 is large in the ordinary sense and small 2-adically; $\tfrac12$ is the reverse — small in the ordinary sense and, having a 2 in its denominator, 2-adically large, of size 2. No ratio is small in every sense at once except 0.

Where two of them overlap, the series converges in both senses — and it converges to the same number, because the partial sums are the same fractions and the formula $1/(1 - r)$ is algebra. With $r = \tfrac23$ the partial sums $1, \tfrac53, \tfrac{19}{9}, \dots$ approach 3 in the ordinary sense, and they approach 3 in the 2-adic sense too, since their differences from 3 are $3 \cdot (\tfrac23)^{n}$, each carrying $n$ factors of 2. Two completely different notions of closeness agree on the limit because there is only one fraction the formula can give.

Why this is more than a curiosity #

The $p$-adic numbers were introduced by Kurt Hensel around 1897, and his motivation was exactly the one this essay has followed: to treat a number as an infinite series in powers of a prime, the way a function is treated as a power series in $x - a$. Under that analogy the prime $p$ plays the role of $x - a$ — small because it vanishes at the point of interest — and a $p$-adic integer is to a whole number what a power series is to a polynomial. Lifting a root one digit at a time is Newton’s method in this setting, and it works for the same reason the geometric series converges: each step’s correction is smaller, $p$-adically, than the last.

The payoff is the local–global principle of Helmut Hasse. To decide whether a quadratic equation has a solution in rational numbers, it is enough to check that it has one in the real numbers and in the $p$-adic numbers for every prime $p$ — and each of those checks is a finite computation, because $p$-adic solutions can be found by lifting. The three-square theorem, whose integer version a rational point on a circle forces, is proved by exactly that route. Every prime gives the rational numbers a different notion of size, and the rational numbers are what all of them see at once.

What the figures cannot show #

The p-adic numbers themselves are not drawn. The figures show whole numbers and fractions, their digits and their distances. The $p$-adic numbers are the completion — every infinite path down the tree, not only those that end in zeros — and most of them are neither whole numbers nor fractions. They cannot be drawn in full any more than the real numbers can, and the tree of depth four is a finite shadow.

The 2-adic digits are computed, not looked up. Each expansion in the table is produced digit by digit — take the remainder modulo 2, subtract, divide by 2 — and then checked by multiplication. The method is Hensel’s lifting in its simplest form, and the table is its output, not a list copied from anywhere.

Convergence is checked on finitely many terms. Each partial-sum figure confirms, with exact arithmetic, that the $n$-th difference carries exactly $n$ factors of the prime, for twenty or fourteen terms. That the pattern continues is the one-line formula for the partial sums, not the figure.

The ordering of real numbers has no counterpart. The 2-adic numbers cannot be arranged in a line consistent with their arithmetic; there is no sense in which $-1$ is “less than” 0 2-adically. The figures that place the partial sums on axes use ordinary coordinates for the ordinary quantity and powers for the 2-adic one, and neither axis is a 2-adic number line, because there is none.

Still open: which numbers have periodic expansions in which systems #

Every rational number has an eventually periodic expansion in every base, in the real sense to the right and in the $p$-adic sense to the left, and nothing else does. The corresponding questions for algebraic numbers are open in both directions. In the real numbers it is not known whether the decimal digits of $\sqrt2$ are normal — whether every block of digits appears with the expected frequency — and the same question for the 2-adic digits of $\sqrt{17}$, which exists because 17 is 1 more than a multiple of 8, is equally open and less studied. It is believed that the digits of every irrational algebraic number look random in every base and every direction, and not a single case has been proved.

There is also a question about sums in several metrics at once. A series of rational numbers may converge in the real numbers and in several $p$-adic fields, to limits that are different numbers in each. Which families of such limits can occur, and what they say about the terms, is the territory of $p$-adic analysis and of the theory of zeta values, where the same series is summed at every prime at once, and much of it is still being mapped.

One formula, many rulers #

The formula $1/(1 - r)$ for the geometric series was never a statement about the real numbers. It is algebra — $(1 - r)(1 + r + \cdots + r^n) = 1 - r^{n+1}$ — together with one analytic fact, that $r^{n+1}$ goes to zero. Change the notion of size and that fact moves to a different set of ratios, and the formula follows it there. The absurd claim that $1 + 2 + 4 + \cdots = -1$ is the formula holding in the arithmetic where 2 is small, and the proof that it holds is the same proof, word for word, that fills the square with halves.

That is the real lesson of the provocation the square left behind. A formula can be right in a system where it looks wrong, because what looks wrong is the notion of size, not the formula. The real numbers are one way of completing the rationals — filling in the limits of sequences that ought to converge — and every prime gives another, each with its own geometric series, its own limits and its own way of deciding what counts as small. Minus one is the sum of the powers of two in exactly one of them, and in that one it could not be anything else.