Let’s start with a joke:
A physicist, a biologist and a mathematician are sitting in a street café one morning watching an empty store on the other side of the street. They see someone unlock the store and go in. Time passes. Someone else goes in. More time passes. Then three people come out.
The physicist says, “Our measurements weren’t accurate.” The biologist says, “The two people who went in must have reproduced.” The mathematician says, “If one more person enters the store, it will be empty.”
Of course the joke here is that the mathematician is holding the absurd belief that there are a negative number of people in the store.1 The joke is built on our shared knowledge that although negative numbers make sense in some contexts, they aren’t sensible in the present context. But the knowledge that negative numbers do make sense in some contexts shouldn’t be taken for granted. Centuries after Chinese and Indian mathematicians figured out how to use negative numbers with comfort and ease, Europeans were still struggling to wrap their minds around the concept. It’s a shame that this story isn’t taught more broadly in the West, and that the real number system we teach to students ends up being viewed by many as a European invention. The truth is more interesting.
(drawing by Ben Orlin: check out https://mathwithbaddrawings.com)
THE JIUZHANG SUANSHU (≤ 263 CE)
Let’s visit China first, starting with a problem that’s over seventeen hundred years old:
Sell 2 cows and 5 sheep to buy 13 pigs: there is a 1000 coin surplus.
Sell 3 cows and 3 pigs to buy 9 sheep: there is exactly enough cash.
Sell 6 sheep and 8 pigs, then buy 5 cows: there is a 600 coin deficit.
Tell me: what is the price of a cow, a sheep and a pig respectively?
This problem comes from a book called the Jiuzhang Suanshu, or Nine Chapters on the Mathematical Art. It’s hard to say just how old it is, since it was an evolving document for hundreds of years, but the problem I quote goes back at least to the year 263 CE, when the mathematician Liu Hui wrote a detailed commentary on the book that was incorporated into subsequent versions. If I write the problem in modern algebraic notation as
2C + 5S −13P = 1000
3C − 9S + 3P = 0
−5C + 6S + 8P = −600
where C, S, and P represent the price of a cow, sheep, and pig respectively, then many of you will recognize the problem as a system of three linear equations in three unknowns.2 The surplus and deficit referred to in the problem correspond to what we call positive and negative numbers. Chinese mathematicians developed methods of solving these problems mechanically using bamboo counting rods; surpluses were represented by red rods and deficits by black rods (in contrast to the reversed color scheme European accountants invented much later).
You may wish to match wits with Liu Hui and tell him the prices of the respective animals.3
By the year 628, Chinese ideas about negative numbers had made their way to India (or been reinvented by Indian mathematicians); Brahmagupta’s treatise Brâhma-sphuta-siddhânta contains the very modern-sounding assertion “The product of a negative and a positive is negative, of two negatives positive, and of two positives positive.”
THE LILAVATI (1150 CE)
Jump ahead half a millennium from Brahmagupta to the work of the 12th century Indian thinker Bhaskara (sometimes called “Bhaskara II” to prevent confusion with the 7th century thinker of the same name). Bhaskara wrote an extremely popular algebra textbook called the Līlāvatī (named after his daughter; the Sanskrit word means “playful one”). Mathematician David Mumford, in his valuable essay “What’s So Baffling About Negative Numbers?” (the main source I’m drawing upon this month) calls our attention to verse 166, in which Bhaskara asked
In a triangle, wherein the sides measure ten and seventeen and the base nine, tell me promptly, expert mathematician, the segments, perpendicular and area.
(Note that where Liu Hui simply said “Tell me”, now Bhaskara spices the command with flattery. Thus does pedagogy advance through the ages.)
Bhaskara derived a formula for problems of this kind, and you can follow along: Suppose c is the base of a triangle, with a and b being the other two sides, and suppose the altitude has length h and divides the base into segments of length x and c−x, as shown below.
The Pythagorean4 theorem applied twice gives us
b2 − x2 = h2 = a2 − (c−x)2 = a2 − c2 + 2cx − x2 ;
cancelling the x2 ‘s we get b2 = a2 − c2 + 2cx, so 2cx = c2 + b2 − a2 , which gives us the handy formula x = (c2 + b2 − a2) / 2c. Plugging in the numbers a = 17, b = 10, and c = 9, we get x = (81 + 100 − 289) / 18 = −108/18 = −6.
Wait what? Negative six?!
Bhaskara was unperturbed by the appearance of a negative number, or should I say, he was nonnonplussed:
[The 6] is negative, that is to say, in the contrary direction. Thus the two segments are found, 6 and 15. From which, both ways too, the perpendicular comes out 8.
The actual picture of Bhaskara’s paradoxical triangle looks like this:
The negativity of x means that our first picture was inapplicable (specifically, the foot of the altitude to the side of length 9 actually falls outside of it). To avoid confusion, we should not think of x as a distance, but rather as a displacement; specifically, x is the displacement of P (the foot of the altitude) from A toward B, and the fact that x is negative corresponds to the fact that in this case, P is reached from A not by traveling 6 units toward B but by traveling 6 units in the opposite direction.
Working with displacement rather than distance can unify what seem to be disparate cases. Sometimes the distance between A and B equals the distance between A and P plus the distance between P and B, and sometimes it doesn’t (depending on the order in which the points A, B, and P fall on a line); but in every case, the displacement from A to B equals the displacement from A to P plus the displacement from P to B.
Mumford argues that what we see in this passage from Bhaskara is not just an interpretation of negative numbers; it’s the first prototype of the modern number line, wherein positive and negative numbers correspond to points on either side of an origin.
LIBER ABACI (1202 CE)
It would be fun to read a Neal Stephenson novel inspired by the life of Leonardo of Pisa, more commonly known by his patronymic Fibonacci5. Keith Devlin in his book The Man of Numbers already goes halfway there, giving us a view of Leonardo as a sort of Steve Jobs figure, with the Hindu-Arabic number system as the thirteenth century equivalent of the iPhone. But a novelist untrammelled by mere historical fact could go much farther.
Leonardo spent years stationed in Algeria working as a shipping agent on behalf of Pisan merchants. During that time he traveled extensively and became familiar with ideas and methods that the Arab world had adapted from the mathematics of China and India. His Liber Abaci was addressed to traders who needed better systems for carrying out computations and (just as importantly) for recording them. In his preface, he wrote:
As my father was a public official away from our homeland in the Bugia customshouse established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle Euclidean geometric art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. If, by chance, something less or more proper or necessary I omitted, your indulgence for me is entreated, as there is no one who is without fault, and in all things is altogether circumspect.
You will note that Leonardo openly acknowledges the nationalist component of his motivation (though “nationalist” is not quite the right word since Italy was not yet a nation). Let us now take a wanton leap from the world of historical fact into the world of the middlebrow thriller, and ask: what if — what if — Leonardo, in addition to working as an agent of individual traders, was secretly an agent of a mercantile cabal that saw commerce as the path forward to Italian economic ascendancy, and saw mathematics as an essential grease for the wheels of Italian commerce? What if Leonardo’s study of the mathematics of the Arab world was really a case of industrial espionage, no less than the British theft of rubber-tree seeds from Brazil in the 1800s? True, his acknowledgment of nationalistic sentiment might seem to be the sort of admission that an agent would avoid, but what better lie is there than an incomplete truth?
One problem with this fictional premise is that one of the most potentially revolutionary ideas of all, the idea of negative numbers, failed to catch on in Italy, in large part because Leonardo didn’t seem to have appreciated just how powerful it was. In the book he did present some financial problems in which the answer is negative, and he correctly interpreted negativity as indicating a debt as opposed to a surplus, but he gave no operating manual for negative numbers. Had he wanted to promulgate their use, he could easily have presented instructions for operating with them. But instead he presented mere isolated examples of negative numbers that had little impact on the practice of his contemporaries.
(Note to Neal Stephenson: The oldest surviving version of Leonardo’s book dates back to 1227, a quarter of a century after the book’s publication. Could it be that the 1202 version did indeed contain instructions on how to work with negative numbers, with examples of their power? Could it be that all copies of the 1202 version of the book were not simply lost, but deliberately collected and destroyed, by Leonardo or by someone else? Of course it would be irresponsible for a serious historian of mathematics to advance such an outrageous suggestion. But you’re a novelist.)
TWO NEW SCIENCES (1638 CE)
It was brash of Galileo to name his last great work Discourses and Mathematical Demonstrations Relating to Two New Sciences. After all, very few people in history have established a whole new science on their own; to claim to establish two of them in one publication smacks of grandiosity. Yet Galileo did indeed lay the foundations of both materials science and ballistics in that single work.
One of Galileo’s key insights was that under the influence of gravity, the downward speed of a falling object increases linearly over time. He also realized that the upward speed of a rising projectile decreases linearly over time. Nowadays we link these observations via the concept of velocity, a quantity that, like the displacement x in Bhaskara’s triangle problem, can be positive or negative, with a change of sign corresponding to a change of direction. When we toss an object into the air, its upward velocity will be positive for a while, then zero for an instant, and then negative afterwards, corresponding to the object’s rise, peak, and fall. Nowadays we would say that upward velocity is a linear function of time throughout the whole process; the moment when the object reaches its peak (with upward velocity momentarily equal to zero) may be dramatic to onlookers but it does not signal a change in the underlying physical phenomenon or in the mathematics needed to describe that phenomenon.
In contrast to Galileo’s theory of projectile motion, Aristotle’s involved a discontinuity in time: a moment at which the projectile, like Wile E. Coyote in Warner Brother cartoons, suddenly realizes that it is governed by the law of gravity and ought to fall. You might have expected that the brash Galileo, eager to overturn the half-baked speculations of pikers like Aristotle, would have trumpeted the way his theory unified the study of rising and falling trajectories and abolished Aristotle’s false discontinuity.
… except that Galileo never quite saw it this way. You will search his book in vain for pictures of full parabolas; you’ll find only half-parabolas, depicting the upward or downward journey of an object but never both together. He considered the problem of the optimal angle at which to fire a gun (note to Stephenson: see, this stuff has applications to warfare as well as commerce), but he avoided drawing or discussing the rise and fall of a projectile in a single analysis (even though the classic parabola of the Greeks was infinite in both directions).
Why did he not unify the two analyses? Because he would have needed negative numbers in order to do so, and these were still regarded with suspicion. A century before Galileo, the mathematician Michael Stifel had published his 1544 Arithmetica which, in a move that prefigured modern scientific notation for big and small numbers, included for the first time the use of negative numbers as exponents. Stifel realized that his use of negative numbers would be considered outré by his contemporaries, and he was careful to call negative numbers “absurd” and “fictitious”, knowing that others would say worse things if he didn’t acknowledge their objections in advance. On the subject of negative exponents, he wrote “Marvelous books could be written, but I myself will refrain and keep my eyes shut.” (Source, anyone?) Stifel never wrote one of those “marvelous” books, and neither did Galileo.
It was interesting for me to learn some of the history of negative numbers, and to see just how backward mid-millennium European mathematicians were, compared to Chinese and Indian mathematicians of an earlier age. My reading made me wonder: Did any Chinese or Indian mathematicians in Leonardo of Pisa’s time, or the centuries that followed, take note of what was going on in Europe? What did they think of it? Was their impression of Western mathematics at all similar to Gandhi’s apocryphal quip about Western civilization (“I think it would be a good idea”)?
The story of negative numbers in China and India reminded me of a parallel story about smallpox: China and India both had longstanding traditions of variolation, the practice of deliberately giving people mild cases of smallpox to prevent them from dying of more serious cases. Imported into Europe by Lady Mary Wortley Montagu in the 1700s, the practice evolved into the technology of vaccination. Is it a coincidence that in both these matters, the Indians and Chinese were ahead of the Europeans?
Both negative numbers and variolation have about them the whiff of paradox. Paradox was not unknown in the Christian West; indeed, reconciling the mortality and divinity of Christ was a major preoccupation of theologians. Ditto for the nature of the Eucharistic Host. But getting the wrong resolution of such a paradox could get you killed. Might it be that paradoxes were more tolerated in other parts of the world, at least for most of the past two thousand years? Could China’s dualistic philosophy, with its acceptance of yin and yang, have made it a more fertile ground for a “dualistic” number system?
Thanks to Sandi Gubin and Ben Orlin.
#1. There can be a number of negative people in a store, but that’s rather different.
#2. Some of you may know how to solve the problem using the method of systematic elimination of variables, aka Gauss-Jordan elimination. It’s worth mentioning that Liu Hui used the same method many centuries earlier.
#3. C = 1200, S = 500, P = 300. I would’ve thought pigs were pricier than sheep, but I’m no farmer, and in any case, I’m not in a position to criticize: I have not always been super-careful about real-world accuracy in my own word problems. A few years ago my students justifiably mocked me for crafting a word problem about the inclusion-exclusion principle that absurdly depicted strawberry ice cream as being roughly as popular as chocolate and vanilla.
#4. The Pythagorean Theorem was known in many cultures long before Pythagoras. Any ideas for what we should call it? The “Mesopotamian Theorem”?
#5. Leonardo is best known today for the number sequence that got named after him, but Fibonacci numbers were known to Indian mathematicians long before Leonardo. Any ideas for what we should call them? “Pingala numbers”?
Keith Devlin, The Man of Numbers, 2011.
Keith Devlin, Finding Fibonacci, 2017.
David Mumford, “What’s So Baffling About Negative Numbers?”.
Jim Propp, “Going Negative, part 1”.
Jim Propp, “Going Negative, part 2”.
Reblogged this on Mathematics Hothouse.
Re fn. 4: I’ve heard, and have started using, “the Right Triangle Theorem” for a^2 + b^2 = c^2.
LikeLiked by 1 person
I like it. Easy to remember. Plus, the theorem was probably discovered independently in many cultures, so singling out one of them seems unfair.
LikeLiked by 1 person
The Chinese know the Pythagorean Theorem as the Gou Gu Theorem — one of my Chinese students told me (I haven’t verified this) that the name comes from some tool that had the shape of a right triangle. Every time it comes up, I ask my students by what name they learned it; as far as I have encountered, every country other than China seems to have learned the European name. (This is in marked contrast to Pascal’s Triangle, for which my students have told me at least four different names.)
LikeLiked by 1 person
Pingback: Going Negative, part 4 |