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The birth pains of Zero

0808

— 11 min read

You’re a Babylonian grain merchant. 

You have 2 talents of grain in one warehouse, none in the second, 5 in the third. 

You need to write that down for a transaction that happens tomorrow when you’re not there; being overseen, for gods sake, by dumbo…

  • your dimmest son-in-law…
  • and this is a whale of client…
  • get this right yaaay…
  • get it wrong…
  • meet thy maker!! 
  • No pressure, then…

Without a placeholder you write “2 5” and that half wit son-in-law of yours doesn’t know if you mean 125 or 7,205 or 2 talents and 5 shekels. 

The ambiguity is not philosophical, or mathematical; without the ‘0’ it’s a fucking law suit waiting to happen…depending on which way the dimwit jumps…

Why, you ask? What’s the problem? 

Put in a fucking zero!!

And there’s the rub; what’s a zero; an absence, a space holder, a presence, a number, what? 

How do you define it? I mean across a global commercial environment of buying, selling across geography and languages; where nobody even understands what’s missing in the notation system, never mind what to fill it with…

Prevailing Counting base

The Babylonian merchants loved base 60, sexagesimal; and it is exceptionally elegant. 

The lowest number wholly divisible by the first 6 numbers; 

  • 1, 2, 3, 4, 5, & 6…also by…
  • 10, 12, 15, 20, 30, 60 (rendering parts of 60 as 6, 5, 4, 3, 2, 1)..
  • Pure symmetry…
  • See Appendix 1 - for a more detailed expansion 

Option 1

The dim wit’s options for deciphering what that doddery old fart’s written down is ‘2no space 5’ that’s 2 x 60 (base in 2nd column, moving left,  from rightmost) + 5 (units in rightmost column)= 125 in base 10…

Option 2

But, if the dimwit reads “2 5” as two digits with a gap representing a missing middle position; that’s 2 in the 60² column, nothing in the 60s column, 5 in the units column = 7,205 in base 10…

Option 3

A talent was 60 shekels in Babylonian weight, so 2 talents and 5 shekels = 125 shekels, the same as Option 1… compelling choice, 2 out of 3…

  • Option 1:  “25” is 2 x 60 (the base) + 5 (units)      = 125 base 10…
  • Option 2:  “2 space 5” is 2 x 3600 + 0 x 60 + 5   = 7,205 base 10…
  • Option 3:  “2 space 5” is 2 talents and 5 shekels = 125 base 10

A range from 125 to 7,205; 

  • while Occam’s razor might incline towards the simplest, 
  • option 3, it’s equal to 
  • option 1; so 
  • 2 out of 3 must be right; 

it may actually slip and cut one’s throat for cheating by a humongous factor of 50…. Ouch!


This absent ‘0’ was an administrative, legal, commercial, engineering and scientific nightmare; 

The written ‘zeroless’ number became a legal instrument. Ambiguity in a number wasn’t an intellectual problem, it was a fraud problem, which made the commercial and legal worlds a serious health hazard of the fatal variety. 

Especially under Hammurabi’s system of laws, the ultimate benchmark for penalties reciprocal to the crime…Miss a space, miss a life…

  • come back Hammy, all is forgiven… rule the world..

The Code of Hammurabi is one of the oldest written legal codes. 282 laws carved into a diorite stele, looted by the French, and now in the Louvre. I bet its owner might have a thing or two to say about that…

And he certainly didn’t write, painfully into diorite (talk about writer’s cramp), 282 laws to govern a couple of chicken farmers exchanging value for profit…

By Hammurabi’s time (1754 BCE) Babylon was already a large urban commercial civilisation with palace economies, temple granaries, tax collection, inter-city/country/nation/peoples trade, credit transactions, and a legal code specifically governing commercial disputes. 

The very first banking system came out of the well-built, weather and vermin-proof, palace granaries issuing receipts for grain deposits and taxing accordingly…a practice mirrored by private palace-standard-built (keeps the pests out and the grain dry) home owners; fore runners of the ’private’ bankers of today’s billionaire world. 

All that without a single, fucking ‘0’ in sight…astonishing…

Imagine trying that on with the tax man… no, no that’s 4 spaces not 5… see my ruler shows 4 spaces… 

just for a larf…try balancing your account without ‘0’..

This was no village market; this was a sophisticated, bureaucratic state apparatus the rival of modern times. 

This system ran on written records; legally mandated written records under Hammurabi’s code; and those written records were forever ambiguous without a zero placeholder.

The scribe wasn’t a bookkeeper keeping personal accounts. The scribe was the legal instrument of the state. His clay tablet was the contract, the receipt, the tax record, and the court evidence all in one.

The placeholder; the symbol that places the significant digits, the leftmost, under the right counting base column..

  • Eg. Do the numbers ‘2  5’ and ‘2     5’ look to have the same value
  • well, that would depend on how many spaces the significant digit ‘2’ is from the rightmost digit ‘5’; assuming spaces are place holders…
  • where’s my ruler…
  • if 3 spaces then  €20,005, 
  • if 4 spaces then €200,005
  • A delta of              €180,000; not exactly a rounding error…
  • If 5 spaces then  €1,800,000; you see where this is going…

Straight to..

Law 5: if a judge makes an error in a case, he pays twelve times the penalty.
Now, would that be 12 x 180,000 or 12 x 1.8 million…

Hammurabi’s code had specific laws about grain storage, contracts, and merchant transactions. Laws 100-107 cover merchant and agent relationships, interest on loans, record keeping obligations.

So the Babylonian merchant writing ambiguous grain tallies without a zero placeholder was operating under a legal system that mandated written commercial records. 

The ‘0’ ambiguity wasn’t just inconvenient, it was downright legally dangerous. A disputed tally of “2 5” could end up before a judge applying Hammurabi’s code.

The first scribe who saw the problem for what it was and invented the placeholder wedge, great unsung hero, solved a legal problem long before it became a mathematical or philosophical curiosity.

The most dangerous number in Babylonian commerce was the one that didn’t exist yet. 

The void. The empty warehouse. 

The nothing that needed a name before it could be safely written down.

Because, once we name something, it bursts into existence…

god, unicorns, money, 0…

we are gods of existence …

Appendix 1 - Sexagesimal on your own time…


The Babylonian merchants loved base 60, sexagesimal; and it is exceptionally elegant. For them, it was a flexible range of whole ‘fractions’ without once have to deal with decimals, since most commercials transactional fractions came from…

1 part of   60 - 60

2 parts of 60 - 30

3 parts of 60 - 20

4 parts of 60 - 15

5 parts of 60 - 12

6 parts of 60 - 10

10 parts of 60 - 6

12 parts of 60 - 5

15 parts of 60 - 4

20 parts of 60 - 3

30 parts of 60 - 2

60 parts of 60 - 1

A pallate to mix and match practically any whole number required…

Any lower, just tell ‘em you don’t do decimals; 

  • in fact, you don’t even know what they are. Babylonian merchants were never fooled by the 10 fingers nature gave us;
  • try dividing 3 parts in 10 into something meaningful;
  • 0.3333333333….. (need I go on?)…
  • Split into 3 equal parts, changes the counting base from 10 to 3…

[Note: the math for the last point, in case it’s not clear; 

Take a heap of grain yaaay big..

Divide by 3 in base 10 = 0.333… (huh,!!- what is 0.3333 of yaaay big?)

Take a heap of grain yaaay big..

Physically divide ‘yaaay big’ into 3 equal parts

The base has changed

Now ‘yaaay big’/3 = 1]

In base 10 the rightmost column (least significant digit)before the decimal is units (10 to the power of 0), then, moving left, 10s (power of 1)=10, then left again, 10x10 (power of 2)=100, then 10x10x10 (power of 3)=1,000 etc…

In base 60 the rightmost column before the decimal is units, then 60s, then 60x60= 3,600, then 60x60x60= 216,000 etc…{a ‘1’ under this column eg 100 in base 60= 216,000 in base 10)…

So elegant indeed that we use it to the present day, just as they did

For time;

  • 60 seconds in a minute, 
  • 60 minutes in an hour
  • 360 degrees in a circle 

For Earth’s rotation

  • 60 arc minutes in a degree
  • 60 arc seconds in an arc minute
  • 1 arc second = 1/3600th of a degree

Nowadays, a ship’s position in latitude and longitude is still expressed in degrees, arc minutes, and arc seconds. Babylonian base 60 is still how we locate ourselves on the planet.

Earth’s rotation: the sky moves 1 degree every 4 minutes or 360 degrees in 24 hours = 15 degrees per hour = 1 degree per 4 minutes. The arc minute / time minute conflation is a common confusion.

In angular measurement there are 60 arc minutes per degree and 60 arc seconds per arc minute, courtesy of the base 60 elegance. But arc minutes and time minutes are different things that share the same name and the same base because both come from Babylon.

Ironic that a system that couldn’t represent zero cleanly gave us the base metric for our 21st century global positioning system - GPS; and it’s not one that has any contenders, after 4,000 years…

The system we use to measure every second of every day. Zero arrived, base 60 survived. They coexist in your watch.​​​​​​​​​​​​​​​​

Appendix 2 - The history of 0


Babylonian System (c. 300 BCE)

Base 60 system (sexagesimal).

Used spaces to show empty places; no symbol for “nothing.” 

Example: 2 5 could mean 2×60 + 5 = 125 or 2×3600 + 5 = 7205 → ambiguous!

Later they added two diagonal wedge marks (n) to mark an empty position;

  • still not a number, just a positional marker…

Mayan System (c. 300 CE)

Base 20 system (vigesimal).

Invented an actual symbol for zero; a shell glyph (n). 

Example: 1 shell 3 = 1×400 + 0×20 + 3 = 403.  

0 acts as a placeholder, beginning to behave like a digit.

Indian System (c. 500–700 CE)

The real leap forward. 

Mathematician Brahmagupta names it śūnya from Sanskrit = void/empty. His text was the Brahmasphutasiddhanta, 628 CE and gives ‘0’ arithmetic rules: 

  • a + 0 = a 
  • a – a = 0 
  • a × 0 = 0 

On division, however 

  • a/0 = 0 (Brahmagupta)
  • a/0 = infinity (Bhaskara II -1150 CE) 
  • The modern position is that a/0 is undefined; not zero, not infinity, not any number.

Brahmagupta’s version rejection

The internally consistent reason for a/0 to be undefined rather than a specific value is as follows:

Division is defined as the inverse of multiplication. 

  • If a/0 = x, then x × 0 = a. 
  • But 0 is 0, so x × 0 = 0, and not a, unless a is also 0…

No value of x satisfies this; so the operation simply has no answer.

The special case of 0/0 is even worse, and it’s called indeterminate rather than undefined. 

Because 

  • 0 × anything = 0, 
  • every number simultaneously satisfies 0/0 = x. 

It’s not that there isn’t an answer, there’s just too damn many of them; bit like the gazillion universes of string theory; faith, not science; in maths, if every answer works, it is catastrophic.


Bhaskara’s Version rejection

Where infinity comes in

Bhaskara wasn’t entirely wrong in spirit. 

As a approaches 0 in the denominator, a/x grows without bound ; i.e.  it tends toward infinity. 

But “tends toward” is not the same as “equals.” 

The limit of 1/x as x→0 is infinity, 

but 1/0 itself remains undefined. 

Calculus papers over this with limits; approaching zero without ever arriving… Zeno’s paradox…

In computing, 

Dividing by zero throws an exception or returns NaN (Not a Number); the machine’s way of saying undefined. 

In physics, 

Equations that produce 1/0 signal a breakdown in the model, not a real infinity; a singularity at the centre of a black hole is the model failing, not reality actually becoming infinite.

So the honest modern answer: division by zero is a hole in arithmetic that we navigate around rather than through.​​​​​​​​​​​​​​​​

  • Zero broke arithmetic at the division step and we still paper over it.
  • Zero becomes both a number and a place marker.

Greek Maths

 The Greeks had sophisticated mathematics; 

Euclid’s geometry;

Pythagoras;

Zeno, but no zero. 

Philosophical resistance: how can nothing be something? 

Aristotle explicitly argued against the void. 

This man has a lot to answer for; from gender bias to piss poor scientific practice; ran every claim like a thought experiment; solvable by logic alone; fuck observation,that just gets in the way of a good theory; a higher IQ Trump…

I mean, this fucker, the greatest thinker of all time declared women to be 

  • defective men, 
  • fewer teeth than men (he was married twice and never counted), 
  • colder blood, obviously never slept next to either on a hot summer’s night; my wife’s a bloody oven…
  • incomplete souls: confession as accusation; another Trump adjacency…

All logic, zero observation. 

The teeth alone would have taken thirty seconds to falsify.

Certainly no Popper, this man. 

I mean, Galileo dropped a heavy and a light weight from the Tower of Pisa simultaneously — Aristotle’s logic said the lighter one falls slower; they landed together…20 seconds of observation… 

like his wives’ teeth… perhaps he couldn’t count…

The Greeks who gave us logic couldn’t accept zero…

  • coz the old ipse dixit fucker, Aristotle said so, 
  • the ethos he garnered was inversely proportional to its justification…

Arabic Transmission → Europe (c. 900–1200 CE)

Arabic scholars call it sifr (empty, khali) from śūnya → sifr → Latin zephirum → zero in English. 

Introduced to Europe by Fibonacci (1202). Example:

205 → 2×100 + 0×10 + 5×1

Enabled the rise of modern positional decimal notation.

Modern Role

As a number: represents the absence of quantity. 

As a cipher: fixes place value in multi-digit numbers. 

As a concept: bridges “nothingness” with “counting”; made algebra, calculus, and computing possible.

Zero is both: a number and a cipher; the simplest idea that changed mathematics forever.

Appendix 3 - The Base Change


The base change pain from 60 to 10 is underappreciated. 

Base 60 to base 10 wasn’t just a notation swap; it was a complete rewiring of how we think about quantity. 

Sixty has beautiful divisibility (2, 3, 4, 5, 6, 10, 12, 15, 20, 30) which is why we still use it for time and angles. 

The Babylonians weren’t stupid; they chose 60 for good reasons. 

Abandoning it for 10 (only divisible by 2 and 5) was a trade of mathematical elegance for simplicity of notation.

But, the when, why, where of this change of just part of our number bases; 

  • we keep Babylon time and spatial positioning; 
  • but we change the base to 10 (divisible by 1, 2 and 5); we got swindled; anyone got the receipt?…

When

Not a single event. A slow replacement across roughly 300 BCE to 1200 CE, region by region.

The Indians were the early adopters of base 10 (decimal)…

  • their number system (1-9 plus zero) was fully decimal by 500-700 CE. 

The Arabs adopted it from the Indians and transmitted it westward. 

Europe only made the switch after Fibonacci’s Liber Abaci in 1202 — and even then it took another 200 years to fully displace Roman numerals in commercial practice. 

Roman numerals survived on clock faces, legal documents, and church inscriptions well into the 15th century.

Why

Two reasons, one practical, one accidental.

Practical: base 10 with positional notation and zero is dramatically easier to compute with on paper or clay. Long multiplication and division in Roman numerals is a nightmare. In positional decimal it’s a learnable algorithm. The abacus (which works in any base) gave way to written calculation precisely because decimal notation made written arithmetic tractable.

Accidental: we have 10 fingers. The Babylonians weren’t just counting fingers, they included finger joints with the thumb, giving 12 per hand, 60 for both combined with the other hand tracking groups of 12. Base 60 required a physical mnemonic. 

An abacus across the backs of human hands. 

Base 10 required nothing but existing as a human, with two hands and 10 fingers…

Where

India first: decimal positional notation originated there. 

Arabia next — al-Khwarizmi’s 9th century texts systematised it. 

Italy as the European entry point as Fibonacci brought it from North Africa where Arab mathematical tradition was dominant. 

Northern Europe last; Germany, France, England resisted longest, clinging to Roman numerals and the abacus.

The irony: the most mathematically elegant base lost to the most anatomically convenient one. 

Ten fingers beat sixty joints every time.


Computational payoff

Without zero,

  • no positional notation,
  • no algorithms (algorithm itself from al-Khwarizmi, the Arab mathematician),
  • no calculus (Newton/Leibniz needed infinitesimals approaching zero),
  • no binary,
  • no computing.
  • no AI… good, bad…

Wool gathering

Zero is the number that doesn’t exist until you need it

  • like god, 
  • like pi, 
  • like probability 1. 







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