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.