# 0 and 1 in Greek mathematics

Following up on an earlier post about zero in Greek mathematics and this timeline of zero, I want to say something more about the role of 0 (zero) and 1 (one) in ancient Greek thought. Unfortunately, some of the discussion on Greek mathematics out there is a bit like this:

## 0 and 1 as quantities

The ancient Greeks could obviously count, and they had bankers, so they understood credits and debts, and the idea of your bank account being empty. However, they had not reached the brilliant insight of Brahmagupta, around 628 AD, that you could multiply a debt (−) and a debt (−) to get a credit (+).

The ancient Greeks had three words for “one” (εἷς = heis, μία = mia, ἑν = hen), depending on gender. So, in the opening line of Plato’s Timaeus, Socrates counts: “One, two, three; but where, my dear Timaeus, is the fourth of those who were yesterday my guests … ? (εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων … ; )

The Greeks had two words for “nothing” or “zero” (μηδέν = mēden and οὐδέν = ouden). So, in the Christian New Testament, in John 21:11, some fisherman count fish and get 153, but in Luke 5:5, Simon Peter says “Master, we toiled all night and took nothing (οὐδὲν)!

## 0 and 1 in calculations

In ordinary (non-positional) Greek numerals, the Greeks used α = 1, ι = 10, and ρ = 100. There was no special symbol for zero. Greek mathematicians, such as Archimedes, wrote numbers out in words when stating a theorem.

Greek astronomers, who performed more complex calculations, used the Babylonian base-60 system. Sexagesimal “digits” from 1 to 59 were written in ordinary Greek numerals, with variations of ō for zero. The overbar was necessary to distinguish ō from the letter ο, which denoted the number 70 (since an overbar was a standard way of indicating abbreviations, it is likely that the symbol ō was an abbreviation for οὐδὲν).

Initially (around 100 AD) the overbar was quite fancy, and it became shorter and simpler over time, eventually disappearing altogether. Here it is in a French edition of Ptolemy’s Almagest of c. 150 AD:

In Greek-influenced Latin astronomical calculations, such as those used by Christians to calculate the date of Easter, “NULLA” or “N” was used for zero as a value. Such calculations date from the third century AD. Here (from Gallica) is part of a beautiful late example from around 700 AD (the calendar of St. Willibrord):

Outside of astronomy, zero does not seem to get mentioned much, although Aristotle, in his Physics (Book 4, Part 8) points out, as if it is a well-known fact, that “there is no ratio of zero (nothing) to a number (οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν),” i.e. that you cannot divide by zero. Here Aristotle may have been ahead of Brahmagupta, who thought that 0/0 = 0.

## 0 and 1 as formal numbers?

We now turn to the formal theory of numbers, in the Elements of Euclid and other works. This is mathematics in a surprisingly modern style, with formal proofs and (more or less) formal definitions. In book VII of the Elements (Definitions 1 & 2), Euclid defines the technical terms μονάς = monas (unit) and ἀριθμὸς = arithmos (number):

1. A monas (unit) is that by virtue of which each of the things that exist is called one (μονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται).
2. An arithmos (number) is a multitude composed of units (ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος).

So 1 is the monas (unit), and the technical definition of arithmos excludes 0 and 1, just as today the technical definition of natural number is taken by some mathematicians to exclude 0. However, in informal Greek language, 1 was still a number, and Greek mathematicians were not at all consistent about excluding 1. It remained a number for the purpose of doing arithmetic. Around 100 AD, for example, Nicomachus of Gerasa (in his Introduction to Arithmetic, Book 1, VIII, 9–12) discusses the powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512 = α, β, δ, η, ιϛ, λβ, ξδ, ρκη, σνϛ, φιβ) and notes that “it is the property of all these terms when they are added together successively to be equal to the next in the series, lacking a monas (συμβέβηκε δὲ πᾱ́σαις ταῖς ἐκθέσεσι συντεθειμέναις σωρηδὸν ἴσαις εἶναι τῷ μετ’ αὐτὰς παρὰ μονάδα).” In the same work (Book 1, XIX, 9), he provides a multiplication table for the numbers 1 through 10:

The issue here is that Euclid was aware of the fundamental theorem of arithmetic, i.e. that every positive integer can be decomposed into a bag (multiset) of prime factors, in no particular order, e.g. 60 = 2×2×3×5 = 2×2×5×3 = 2×5×2×3 = 5×2×2×3 = 5×2×3×2 = 2×5×3×2 = 2×3×5×2 = 2×3×2×5 = 3×2×2×5 = 3×2×5×2 = 3×5×2×2 = 5×3×2×2.

Euclid proves most of this theorem in propositions 30, 31 and 32 of his Book VII and proposition 14 of his Book IX. The number 0 is obviously excluded from consideration here, and the number 1 is special because it represents the empty bag (even today we recognise that 1 is a special case, because it is not a prime number, and it is not composed of prime factors either – although, as late as a century ago, there were mathematicians who called 1 prime, which causes all kinds of problems):

• If two numbers (arithmoi) by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers (ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει) – i.e. if a prime p divides ab, then it divides a or b or both
• Any composite number is measured by some prime number (ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – i.e. it has a prime factor
• Any number (arithmos) either is prime or is measured by some prime number (ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – this would not be true for 1
• If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it (ἐὰν ἐλάχιστος ἀριθμὸς ὑπὸ πρώτων ἀριθμῶν μετρῆται, ὑπ᾽ οὐδενὸς ἄλλου πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν ἐξ ἀρχῆς μετρούντων) – this is a partial expression of the uniqueness of prime factorisation

The special property of 1, the monas or unit, was sometimes expressed (e.g. by Nicomachus of Gerasa) by saying that it is the “beginning of arithmoi … but not itself an arithmos.” As we have already seen, nobody was consistent about this, and there was, of course, no problem in doing arithmetic with 1. Everybody agreed that 1 + 2 + 3 + 4 = 10. In modern mathematics, we would avoid problems by saying that natural numbers are produced using the successor function S, and distinguish that function from the number S(0) = 1.

The words monas and arithmos occur in other Greek writers, not always in the Euclidean technical sense. For example, in a discussion of causes and properties in the Phaedo (105c), Plato tells us that “if you ask what causes an arithmos to be odd, I shall not say oddness, but the monas (οὐδ᾽ ᾧ ἂν ἀριθμῷ τί ἐγγένηται περιττὸς ἔσται, οὐκ ἐρῶ ᾧ ἂν περιττότης, ἀλλ᾽ ᾧ ἂν μονάς).” Aristotle, in his Metaphysics, spends some time on the philosophical question of what the monas really is.

In general, the ancient Greeks seem to have had quite a sophisticated understanding of 0 and 1, though hampered by poor vocabulary and a lack of good symbols. Outside of applied mathematics and astronomy, they mostly worked with what we would call the multiplicative group of the positive rational numbers. What they were missing was any awareness of negative numbers as mathematical (not just financial) concepts. That had to wait until Brahmagupta, and when it came, 0 suddenly became a whole lot more interesting, because it eventually became possible to define more advanced mathematical concepts like fields.

# The history of zero: an infographic

Following up on an earlier post about Zero in Greek mathematics, here is a timeline for the use of zero in Europe (click to zoom). I have used images of, or quotes from, primary sources where possible (reliably dated Indian primary sources are much harder to find than Greek ones, unfortunately).

Chinese uses of zero are probably also derived from the Greeks, but Mayan uses are clearly independent.

# Dante’s Heaven

In previous posts (Inferno, Purgatorio, Paradiso), I have mentioned the scientific content of Dante’s incredible theological poem, the Divine Comedy. Above, just for fun, is a chart of Heaven (the Solar System) in his Paradiso. Notice the sphere of fire which was believed to surround the Earth.

# Historical detective fiction

I have been reflecting on the way in which historical detective fiction continues to surprise me. The historical setting helps determine motive in this genre – what made the criminal commit the crime? The criminal’s motive will depend, in part, on the society in which he or she lives, and so the historical setting is generally well described. Indeed, an interesting setting is a bonus feature of fiction set in foreign times and/or places. However, the details of the setting can be surprising to someone who is not an expert in history. Here are four detective books that I learned something from:

First is Murder Must Advertise by Dorothy L. Sayers. Dorothy Sayers worked for many years in an advertising agency, and wrote the famous jingle for the Guinness Toucan campaign (see below). Her depiction of 1930s advertising in Murder Must Advertise is surprisingly modern in mindset, though not in technology:

“Of course, there is some truth in advertising. There’s yeast in bread, but you can’t make bread with yeast alone. Truth in advertising … is like leaven, which a woman hid in three measures of meal. It provides a suitable quantity of gas, with which to blow out a mass of crude misrepresentation into a form that the public can swallow.”

What is not modern, though, is the use of child labour:

“Well, I suppose I’m naturally inquisitive. I always like to know about people. About the office-boys, for instance. They do physical jerks on the roof, don’t they? Is that the only time they’re allowed on the roof?”
“They’d better not let the Sergeant catch ‘em up there in office-hours. Why?”
“I just wondered. They’re a mischievous lot, I expect; boys always are. I like ‘em. What’s the name of the red-headed one? He looks a snappy lad.”
“That’s Joe – they call him Ginger, of course. What’s he been doing?”

By the 1930s, the various Factory Acts and Education Acts had stopped very young children from working, and kept them in school, but it was still common to start work at 14, which is what I imagine Ginger’s age was.

The Marcus Didius Falco series (1989–2010) by Lindsey Davis begins with The Silver Pigs, and is set in the Roman Empire from 70 to 77 AD, during the reign of Vespasian (whose head is on the aureus below). I was surprised at how modern the Romans were in various ways, although Lindsey Davis may be over-emphasising this a little.

The Matthew Shardlake series (2003–2018) by C. J. Sansom begins with Dissolution, and is set between 1537 and 1549, in the reigns of Henry VIII and (for the last book) of his son Edward VI. Shardlake is not only a detective, but a lawyer, with an office in Lincoln’s Inn (below). The later books especially have a lot to say about his legal practice. Once again I was surprised by how modern it was, and the UCLA Law Review has a lengthy discussion of Shardlake’s professionalism.

The Mrs. Meade series (2012–2022), by indie author Elisabeth Grace Foley, is a set of short fun Western mysteries (Foley also writes a variety of other fiction). Her The American Pony is the latest in the series, and stars an English baronet in turn-of-the-century Colorado. I was surprised to discover that English gentry in the old West (the Daily Mail calls them “toffs”) were a real thing, one of the most famous being Moreton Frewen (below), who was forced strongly encouraged to emigrate far from most of his kith and kin in 1878.

In short, detective fiction is not only fun, it can also be educational. I could have listed any number of other authors of the genre that have surprised me in similar ways.

# Looking back: 2001

The 1968 film 2001: A Space Odyssey suggested that we would have extensive space flight in 2001. That turned out not to be the case. What we did get was the September 11 attacks on the USA and the military conflicts which followed. Nevertheless, NASA commemorated the film with the 2001 Mars Odyssey orbiter.

Films of 2000 included the superb The Lord of the Rings: The Fellowship of the Ring, several good animated films (including Monsters, Inc., Shrek, and Hayao Miyazaki’s Spirited Away), the wonderful French film Amélie, some war movies (Enemy at the Gates was good, but Black Hawk Down distorted the book too much for my taste), the first Harry Potter movie, and an award-winning biographical film about the mathematician John Nash.

In books, Connie Willis published Passage, one of my favourite science fiction novels, while Ian Stewart explained some sophisticated mathematics simply in Flatterland.

Saul Kripke (belatedly) received the Rolf Schock Prize in Logic and Philosophy for his work on Kripke semantics, while Ole-Johan Dahl and Kristen Nygaard (also belatedly) received the Turing Award for their work on object-oriented programming languages (both these pioneers of computing died the following year).

The year 2001 also saw the completion of the Cathedral of Saint Gregory the Illuminator in Armenia, which I have sadly never visited.

In this series: 1978, 1980, 1982, 1984, 1987, 1989, 1991, 1994, 2000, 2001, 2004, 2006, 2009.

# Houston, we have a problem

Some years ago, I posted the chart above, inspired by a classic XKCD cartoon. The infographic above shows the year of publication and of setting for several novels, plays, and films.

They fall into four groups. The top (white) section is literature set in our future. The upper grey section contains obsolete predictions – literature (like the book 1984) set in the future when it was written, but now set in our past. The centre grey section contains what XKCD calls “former period pieces” – literature (like Shakespeare’s Richard III) set in the past, but written closer to the setting than to our day. He points out that modern audiences may not realise “which parts were supposed to sound old.” The lower grey section contains literature (like Ivanhoe) set in the more distant past.

The movie Apollo 13 has now joined the “former period piece” category. Released in 1995, it described an event of 1970, 25 years in the past. But the ill-fated Apollo 13 mission of 11–17 April 1970 is now 51 years in the past; the movie is closer to the event than it is to us (although the phrase “Houston, we have a problem” – in real life, “Houston, we’ve had a problem” – has become part of the English language).

The image shows the real-life Apollo 13 Service Module, crippled by an explosion (left), together with a poster for the 1995 movie (right). Maybe it’s time to watch it again?

# Cistercian numerals

Somebody recently pointed me at Cistercian numerals (above), an interesting base-10 numeral system used by Cistercian monks in the Low Countries and northern France from the 1200s to the 1500s (that is, after the Liber Abaci introduced Hindu-Arabic numerals to Europe, but apparently based on an older English system, not on that).

Most extant uses of the system relate to dates and item numbering, rather than arithmetic. This online conversion tool will let you experiment with the system. It is interesting to note the relationships 5 = 4 + 1, 7 = 6 + 1, 8 = 6 + 2, and 9 = 8 + 1 = 7 + 2 = 6 + 2 + 1.

# The Santa Fe Trail #4

NPS map of the Santa Fe Trail in 1871 (click to zoom; more maps here)

The American Solar Challenge is on again in 2021, and includes a road race along the Santa Fe Trail on 4–7 August, from Independence, MO to Santa Fe, NM (exact route still to be decided).

To get myself in the mood, I’ve been reading Land of Enchantment, the memoirs of Marion Sloan Russell, who travelled the Santa Fe Trail multiple times. After marrying, she was an “army wife” for some time, before setting up a trading post beside the Trail. In 1871, she moved to a ranch in the mountains west of Trinidad, CO, where her husband was murdered during the Colfax County War. Towards the end of her life she visited many important sites along the Trail. They were already falling into ruin:

At Fort Union I found crumbling walls and tottering chimneys. Here and there a tottering adobe wall where once a mighty howitzer had stood. Great rooms stood roofless, their whitewashed walls open to the sky. Wild gourd vines grew inside the officers’ quarters. Rabbits scurried before my questing feet. The little guard house alone stood intact, mute witness of the punishment inflicted there. The Stars and Stripes was gone. Among a heap of rubble I found the ruins of the little chapel where I had stood—a demure, little bride in a velvet cape—and heard a preacher say, ‘That which God hath joined together let no man put asunder.’

Marion Sloan Russell died in 1936 (aged 92) after being struck by a car in Trinidad, CO. She is buried in Stonewall Cemetery.

Fort Union in 2006 (credit: Scott; click to zoom)

Other posts in this series: Santa Fe Trail #1, Santa Fe Trail #2, Santa Fe Trail #3, Santa Fe Trail #4.