I recently read *The Nothing That Is: A Natural History of Zero* by Robert M. Kaplan. Zero is an important concept in mathematics. But where did it come from?

## The Babylonian zero

From around 2000 BC, the Babylonians used a positional number system with base 60. Initially a space was used to represent zero. Vertical wedges mean 1, and chevrons mean 10:

This number (which we can write as 2 ; 0 ; 13) means 2 × 3600 + 0 × 60 + 13 = 7213. Four thousand years later, we still use the same system when dealing with angles or with time: 2 hours, no minutes, and 13 seconds is 7213 seconds.

Later, the Babylonians introduced a variety of explicit symbols for zero. By 400 BC, a pair of angled wedges was used:

The Babylonians were happy to move the decimal point (actually, “sexagesimal point”) forwards and backwards to facilitate calculation. The number ½, for example, was treated the same as 30 (which is half of 60). In much the same way, 20^{th} century users of the slide rule treated 50, 5, and 0.5 as the same number. What is 0.5 ÷ 20? The calculation is done as 5 ÷ 2 = 2.5. Only at the end do you think about where the decimal point should go (0.025).

## Greek mathematics in words

Kaplan says about zero that “the Greeks had no word for it.” Is that true?

Much of Greek mathematics was done in words. For example, the famous Proposition 3 in the *Measurement of a Circle* (Κύκλου μέτρησις) by Archimedes reads:

Παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστί, καὶ ἔτι ὑπερέχει ἐλάσσονι μὲν ἤ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις.

Phonetically, that is:

*Pantos kuklou hē perimetros tēs diametrou triplasiōn esti, kai eti huperechei elassoni men ē hebdomō merei tēs diametrou, meizoni de ē deka hebdomēkostomonois.*

Or, in English:

The perimeter of every circle is triple the diameter plus an amount less than one seventh of the diameter and greater than ten seventy-firsts.

In modern notation, we would express that far more briefly as ^{10}/_{71} < π − 3 < ^{1}/_{7} or 3.141 < π < 3.143.

The Greek words for zero were the two words for “nothing” – μηδέν (*mēden*) and οὐδέν (*ouden*). Around 100 AD, Nicomachus of Gerasa (Gerasa is now the city of Jerash, Jordan), wrote in his *Introduction to Arithmetic* (Book 2, VI, 3) that:

οὐδέν οὐδενί συντεθὲν … οὐδέν ποιεῖ (*ouden oudeni suntethen … ouden poiei*)

That is, zero (nothing) can be added:

nothing and nothing, added together, … make nothing

However, we cannot **divide** by zero. Aristotle, in Book 4, Lectio 12 of his *Physics* tells us that:

οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν (*oude to mēden pros arithmon*)

That is, ^{1}/_{0}, ^{2}/_{0}, and so forth make no sense:

there is no ratio of zero (nothing) to a number

If we view arithmetic primarily as a game of multiplying, dividing, taking ratios, and finding prime factors, then poor old zero really does have to sit on the sidelines (in modern terms, zero is not part of a multiplicative group).

## Greek calculation

For business calculations, surveying, numerical tables, and most other mathematical calculations (e.g. the proof of Archimedes’ Proposition 3), the Greeks used a non-positional decimal system, based on 24 letters and 3 obsolete letters. In its later form, this was as follows:

Units | Tens | Hundreds |
---|---|---|

α = 1 | ι = 10 | ρ = 100 |

β = 2 | κ = 20 | σ = 200 |

γ = 3 | λ = 30 | τ = 300 |

δ = 4 | μ = 40 | υ = 400 |

ε = 5 | ν = 50 | φ = 500 |

ϛ (stigma) = 6 | ξ = 60 | χ = 600 |

ζ = 7 | ο = 70 | ψ = 700 |

η = 8 | π = 80 | ω = 800 |

θ = 9 | ϙ (koppa) = 90 | ϡ (sampi) = 900 |

For users of R:

```
to.greek.digits <- function (v) { # v is a vector of numbers
if (any(v < 1 | v > 999)) stop("Can only do Greek digits for 1..999")
else {
s <- intToUtf8(c(0x3b1:0x3b5,0x3db,0x3b6:0x3c0,0x3d9,0x3c1,0x3c3:0x3c9,0x3e1))
greek <- strsplit(s, "", fixed=TRUE)[[1]]
d <- function(i, power=1) { if (i == 0) "" else greek[i + (power - 1) * 9] }
f <- function(x) { paste0(d(x %/% 100, 3), d((x %/% 10) %% 10, 2), d(x %% 10)) }
sapply(v, f)
}
}
```

For example, the “number of the beast” (666) as written in Byzantine manuscripts of the Bible is χξϛ (older manuscripts spell the number out in words: ἑξακόσιοι ἑξήκοντα ἕξ = *hexakosioi hexēkonta hex*).

This Greek system of numerals did not include zero – but then again, it was used in situations where zero was not needed.

## Greek geometry

Most of Greek mathematics was geometric in nature, rather than based on calculation. For example, the famous Pythagorean Theorem tells us that the areas of two squares add up to give the area of a third.

In geometry, zero was represented as a line of zero length (i.e. a point) or as a rectangle of zero area (i.e. a line). This is implicit in Euclid’s first two definitions (σημεῖόν ἐστιν, οὗ μέρος οὐθέν = a point is that which has no part; γραμμὴ δὲ μῆκος ἀπλατές = a line is breadthless length).

In the Pythagorean Theorem, lines are multiplied by themselves to give areas, and the sum of the two smaller areas gives the third (image: Ntozis)

## Graeco-Babylonian mathematics

In astronomy, the Greeks continued to use the Babylonian sexagesimal system (much as we do today, with our “degrees, minutes, and seconds”). Numbers were written using the alphabetic system described above, and at the time of Ptolemy, zero was written like this (appearing in numerous papyri from 100 AD onwards, with occasional variations):

For example, 7213 seconds would be β ō ιγ = 2 0 13 (for another example, see the image below). The circle here may be an abbreviation for οὐδέν = nothing (just as early Christian Easter calculations used N for Nulla to mean zero). The overbar is necessary to distinguish ō from ο = 70 (it also resembles the overbars used in sacred abbreviations).

This use of a circle to mean zero was passed on to the Arabs and to India, which means that our modern symbol 0 is, in fact, Graeco-Babylonian in origin (the contribution of Indian mathematicians such as Brahmagupta was not the introduction of zero, but the theory of negative numbers). I had not realised this before; from now on I will say *ouden* every time I read “zero.”

Part of a table from a French edition of Ptolemy’s

*Almagest*of c. 150 AD. For the angles

*x*= ½°, 1°, and 1½°, the table shows 120 sin(

*x*/2). The (sexagesimal) values, in the columns headed ΕΥΘΕΙΩΝ, are ō λα κε = 0 31 25 = 0.5236, α β ν = 1 2 50 = 1.0472, and α λδ ιε = 1 34 15 = 1.5708. The columns on the right are an aid to interpolation. Notice that zero occurs six times.

I had always thought that it was a bit absurd to assert that the Greeks/Phoenicians and other trading cultures had no concept of zero, but somehow managed to have concepts of debts and credits. How did one know when the debt was paid?

This is a very interesting post – thanks for this.

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Brahmagupta was the first one to give all rules regarding operations in a zero and only inn india zero was used in at end of digit, the law of signs was developed here hence all the rules arose in india

The rules for multiplying signed numbers were indeed discovered in India. On the other points, you are incorrect: see the blog post.

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