# Zero

Created | Updated Dec 23, 2012

We all take zero for granted. It occupies such a natural position in our set of numbers that it is difficult to imagine a time when it wasn't there. But for many centuries that was the case; zero is a relatively recent addition. It's a kind of paradox - it is something to represent nothing, and therein lies the difficulty with which the ancient civilisations viewed it. They were used to natural numbers - numbers you can count - for trading, measuring and the like. There seemed little use for a symbol to represent a quantity that you couldn't buy or sell, or a length of wood that you couldn't cut.

### A Brief History

We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.

- Albert Einstein

A symbol of sorts had been around since 3000 BC, in Sumerian and then Babylonian number systems, but the earliest-known representation of zero as we would recognise it today was devised by the Greek mathematician Ptolemy in the 2nd Century AD. He used the Greek letter omicron Ο to denote zero in a Sumerian base-60 number system - for many centuries before, a slanted double wedge symbol had been used. Zero did not appear in most other number systems of the time, being most notably absent from Roman numerals. The Maya civilisation in Central America later used a shell symbol to represent zero in their independently-developed base-20 system.

After the Sumerian number system had fallen into disuse, it fell to the Indian mathematicians of the 7th Century^{1} to invent a word to represent an absence of a value in their place-value decimal number system. It was important for them to be able to tell apart numbers like 42, 402 and 420. Their word became a dot, and then took a more rounded shape which we would recognise today. We know all this from their writings, but the earliest known representation of the symbol is on a stone tablet of approximately 876 AD.

#### Division by Zero Error

The Indians cannot have known how useful and long-lasting an invention it was going to be. In the early days, however, zero defied many of their rules of arithmetic. Division by zero was the biggest problem, and confused some of the great mathematicians of the time. In the 7th Century, Brahmagupta believed that zero divided by zero was zero, something which we would disagree with today. He wouldn't attempt to divide zero by other numbers, or other numbers by zero, leaving them as fractions like 0/6 or 3/0, but he did correctly speculate that the first of these equated to zero. Two centuries later, the mathematician Mahariva incorrectly believed that fractions like 5/0 equated to five. In the 12th Century, Bhaskara announced that if you were to divide by zero you would get a number which was 'as infinite as the god Vishnu'.

The Indian decimal system eventally spread to China and via the Middle East into Europe, and our modern way of explaining the division problem was largely developed by the German mathematician Georg Cantor in the late 19th Century. We tend to say that division by zero is undefined, and where we need to define mathematical curves which include this division problem at some point, we investigate what happens as the curve approaches zero from either side, and sometimes we can show that it is continuously defined at that point^{2}.

### Some Mathematical Properties

Zero is fundamental to some of our axioms, or laws of mathematics. Anything added to zero is unchanged, and anything multiplied by zero is zero. It's the only number having these properties.

You can raise a number to the power of zero, but the result is always one, whatever the number you started with. Zero to the power of zero is difficult to define, but the result is usually taken to be one.

Zero occupies the no-man's-land between positive and negative numbers. It's the only number for which there isn't a negative counterpart; minus zero is equal to zero - the only number with that property, although this Researcher can confirm that this odd value has occasionally appeared on his car's digital temperature display^{3}.

In our base-10 decimal system, all numbers which are multiples of 10 end in a zero. This is the same for all bases; numbers in base seven which end in a zero are multiples of seven. In the binary (base two) system, it's one of only two symbols, 1 and 0. For this reason, it has a meaning of 'false' or 'off' in logic, where the symbol 1 means 'true' or 'on'.

The presence of zeroes at the end of a number often indicates some importance or recognition - we tend to celebrate anniversaries and milestones based on these 'round numbers'.

Vectors are mathematical quantities with an in-built direction, like a car driving 25mph northwards or a rocket at 100m above the ground. The zero vector has no size, and so it is the only vector which cannot have a direction. If a car is at rest, its direction is immaterial. We often write the zero vector as a bold or underlined zero. It's an important concept in the branch of mathematics known as statics. According to Sir Isaac Newton's laws of motion, if a particle is at rest, then adding together all the forces acting upon it will result in the zero vector. If a rigid body is at rest, then all the torques (turning forces) acting about any point will also add together to result in the zero vector.

Whereas other natural numbers have a corresponding order - first, second, third, etc - there is no order associated with zero. Occasionally the term zeroth has been used to describe the order of a newly-discovered item which logically precedes the one previously described as the first. This is true of the Zeroth Law of Thermodynamics. Bruckner's Symphony No 0 was so numbered for the same reason.

As stated earlier, Ptolemy assigned the Greek letter omicron to zero, and it still looks somewhat like the letter O in our alphabet. This has been a perennial frustration for computer programmers and indeed any of us who have to deal with alphanumeric codes. In some fonts, the zero symbol has been depicted with a line running through it^{4}, although in many proportional fonts used in Graphical User Interfaces we can differentiate zero from an upper case O by it being somewhat thinner.

### Zero - the Word

The word zero came to the English language by the same route as the decimal number system. It derives from the Arabic *sifr*, as does the word cipher; indeed cipher, or cypher, is sometimes used to mean the zero symbol.

What does it mean in terms of quantities which we can count? Often it has a special meaning, and we would employ a different way of expressing this in words. Where other numbers indicate a presence, zero is an absence. A zero volume could be described as 'empty'. Two points which are zero distance apart are 'collocated' or 'in the same place'. A vehicle travelling at zero speed is 'at rest'. Something at zero height could be 'at ground level', or 'at sea-level', etc.

These days, we often use it emphatically; a tough policing policy could be described as 'zero tolerance', and we may have 'zero confidence' in something. We 'zero in on' our target to stress our precision and resolve.

Zero is an origin - a point of reference, a baseline, a starting point from which we start to count or measure in quantifiable numbers. A number of special meanings have arisen as the word has been applied to different situations.

### Special Meanings

Neither a borrower nor a lender be.

- Shakespeare (Hamlet)

If your bank statement totals to zero, then you are financially balanced - neither in the black nor in the red. One other place where zero doesn't fall into that colour pattern is on the roulette wheel, where it's green; it's the additional number which provides the house with its bias. It's neither odd nor even in roulette, but in mathematics, its parity is generally agreed to be even. When you divide it by two there isn't a remainder of one - the 'odd' one left over which gives its name to that particular numeric partition.

On the temperature scale, zero degrees Celsius denotes the freezing point of water. Zero degrees on Fahrenheit's scale was the temperature of a particular, but unspecified, salt-ice mixture. Scientists have also defined 'absolute zero' - the ultra-low temperature at which a substance has no heat energy, equal to -273.15°C or –459.67°F on those scales, or zero on the Kelvin scale, written as 0K.

In geographic location, zero degrees latitude is defined as any point on the equator, and zero degrees longitude is any point on the Greenwich meridian.

In time, zero hour could be the time of the military attack, or the rocket launch. The familiar NASA countdown often ends at this point with the words 'we have lift off'. It also has a special meaning in nuclear disarmament; the 'zero option' is an agreement to limit or abandon short-range missiles, whereas the 'zero-zero option' extends this to those with intermediate range. One further military term is 'ground zero' - the target for a nuclear strike. As Ground Zero, this term is also used for the site of the World Trade Centre in New York, destroyed by Islamic terrorists in suicide plane attacks on 11 September, 2001. Also implying a zero value is the term 'point-blank', indicating a shot or missile which is fired at a target from very close range. We also use this term metaphorically to mean something which is done without explanation or qualification.

Another unit of time is the calendar year, but this curiously lacks a zero value. Our baseline is the year 1 AD, before which came year 1 BC.

The murky world of fashion has recently spawned a 'size zero' - a slimming target which can seriously damage your health.

### Alternative Forms

Of course, English wouldn't be English without more than one way to say something, and with zero we have a fine set of synonyms:

#### Nought

It is naught, it is naught, sayeth the buyer, but when he is gone his way, then he boasteth.

- The Bible (Proverbs 20:14)

Nought, or naught, derives from from the Anglo Saxon *noht*, from *ne* (not) + *wiht* (whit - a very small particle). It's used in the game 'noughts and crosses' (aka Tic Tac Toe), and is also used to describe the 2000-2009 decade - the Noughties.

Nowt is 19th Century English dialect for 'naught'.

#### Nil

Deriving from the Latin *nihil*, nil was famously used by the Roman poet Horace in the phrase *nil desperandum* (Odes I: vii), meaning 'never despair'. This was taken and mocked in the 20th Century as 'nil carborundum', or 'illegitimi nil carborundum' (don't let the, er, illegitimates grind you down), used most notably by World War II General 'Vinegar Joe' Stillwell, commander of US troops in China.

Nil is commonly used to indicate a sporting score of zero. In football, a nil-nil draw isn't much of a spectacle for the watching fans, but 'keeping a clean sheet' is the aim of every team's goalkeeper. Zero is of course the number of points you will be awarded if you lose a league game.

While we're on sport, the cricketing 'duck', indicating a batsman's zero score, was originally 'duck's egg', from the shape of a zero. 'Love' in tennis may derive from the French *l'oeuf* for the same reason, but the OED believes that it most likely comes from the phrase 'to play for love or money', first recorded in 1742 in Hoyle's *'A Short Treatise on the Game of Whist'*. Another cricketing zero term is 'dot ball' - a delivery on which the batsman isn't out, yet doesn't score runs of any kind. The dot is the shorthand way of recording this in a cricket scorebook.

#### Slang Terms

Sticks nix hick pix.(On the lack of enthusiasm for farm dramas among rural populations).

- Headline inVariety, 17 July, 1935.

**Nix** is US slang deriving from the German *nichts*^{5}, first used in the US in the 19th Century, but spelled in the previous century as 'nicks'.

**Zilch** is 1960s' slang, an amalgam of 'zero', 'nil' and the Yiddish *nich*.

**Zip** is 20th Century US slang, as is nada, which derives from the Spanish.

Finally, one other phrase which has entered the English language is *nul points*, pronounced as per the French. Used to ridicule a zero score, this phrase originates from the seemingly interminable scoring phase of the Eurovision Song Contest. While we're discussing European forms, the French would more commonly say *zéro*, and other common European representations are the German *Null*, the Dutch *nul* and the Spanish *cero*. Italians use *zero*.

^{1}In his book

*What the Past Did For Us*, Adam Hart-Davis mentions an earlier text written by Aryhabata in 499.

^{2}An example is the function sine(x) divided by x. This approaches a value of 1 as x approaches zero from either side.

^{3}This is a 'feature' of the car's computer, rather than any insight into mathematical number theory.

^{4}In the early days of computing, IBM chose to put the slash through the letter O!

^{5}Although Nix itself is also used in German, and niks in Dutch.