Wherefrom thou understandest that 1 is no number, but it is a generatrix, beginning and foundation for all other numbers.
– Jacob Kobel, German mathematician, in 1537.
There's a fundamental paradox with the number one - is it a number? When we count, it's the first number we register, yet we reserve the description 'number' for items in plural, whereas one is singular. If I said I was going to give you a number of sweets, then you might be disappointed if I gave you only one.
The Ancient Greeks would have agreed with you - they saw one as the indivisible unit or monad from which all other numbers were created, and they got quite deeply philosophical about it. Euclid said that numbers were composed from units, so therefore the unit itself could not be a number. If you consider, say, six apples, then an apple, or one apple is the unit in which you are counting, and we recognise this in the alternative term for one: unity.
Mathematics for One
It also has some other interesting, if trivial, mathematical properties. You can multiply any number by one, divide it by one, or raise it to the power of one, and the number is unchanged - the only number for which this is the case. One is a factor of all other integers (whole numbers) - they can all be exactly divided by one. As multiplication by one leaves a number unchanged, we don't bother to use it as a multiplier (or coefficient) in algebra - we say 2x or 3x, but not 1x: we'd just write x.
It's the only non-zero number for which addition is a more powerful operation than multiplication. Adding one yields a greater number than multiplying by one.
It's the first odd number, but here again, the Greeks were so much in awe of its fundamental properties, that they felt unable to categorise it, saying it was both odd and even.
When you take the logarithm of one, you get zero, and this is true for logarithms in any base. This unusual property derives from the fact that any number raised to the power of zero is one.
When we talk about fractions, we mean fractions with respect to one. We say ½ is 'one half', but half of what? Of one, of course. In this sense it's the most fundamental number we have.
Finally, it's not only the first natural number, but one is the first square number, the first cubic number, the first triangular number, the first Fibonacci number, in fact it's the first number which we count in many different number systems and series which we have created.
There has been debate over whether one is a prime number. Many of us learned at school that a prime number is an integer which is exactly divisible by only itself and one. Six is divisible by two and three as well, so it's not a prime number. Seven is divisible by only seven and one, so it is prime. But then one would also be a prime number by this definition.
In fact, mathematicians have argued for many years over exactly this. The trouble is that if one is a prime number, then it breaks the rules - specifically the 'fundamental theorem of arithmetic'. This theorem, attributed to the Greek mathematician Euclid, states that there is only one way to represent an integer as the product (multiple) of prime numbers. So 42, for example, is represented as the product of two, three and seven - all prime numbers: 2 × 3 × 7 = 42.
The trouble is, if you include one as a prime number, then 42 can also be represented as the product of one, two, three and seven: 1 × 2 × 3 × 7 = 42 - so it's not the only way.
The Swiss/German mathematician Euler believed that one was not a prime number, for the following reason. If you take the prime factors of a prime number - let's call it p - then these add up to p + 1 (for example, the factors of 7 are 1 and 7, and these, obviously, add up to 7 + 1). Now, he argued, if one itself is a prime number, then its factors would also add up to p + 1, but they don't. There is only one (1) and it adds up to just p.
Today, mathematicians get around this by treating one as a special case, and not a prime number. They say that prime numbers have to be greater than one, or they say they must have exactly one positive divisor other than one. It's a fiddle, but one which seems to make the wheels of mathematical number theory run more smoothly.
So, what's it like to be one? There are a number of distinct ideas associated with the concept of oneness, depending on the context.
Tous pour un, un pour tous. (All for one, one for all.)
Dumas - Les Trois Mousquetaires (1844)
You can combine things into one entity - bringing them together either in space or in purpose, and this is what many understand by the word 'unity'. We have trade unions, workers united in a single cause and voice. Their decisions and votes are invariably 'unanimous'.
There is also a togetherness in terms of time rather than space - perhaps we would call this 'harmony', where we are at one in time, or acting simultaneously with others. Harmony also carries the sense that the whole can be greater than the sum of its parts.
As fractions are considered with respect to one, one itself is the whole. We use terms like 100%, just a shorthand way of writing the fraction 100/100. No part of the whole is missing, it is complete. Complementary medicine has spawned the idea of holistics (from the Greek holos - whole) in terms of a treatment which considers every aspect of a person's well-being.
This is a sense of one being apart or separated from others. We sometimes seek to distinguish ourselves by this, and have the word 'unique' to describe this state. We also use the word 'only', but this is just the adverb for the number one - in fact we used to spell it 'onely' in centuries past. Similarly we might describe ourselves as alone or lonely, which also derive from 'one'. Single derives from a different Latin word, singulus, and we commonly use this in a sense of an incomplete one, for example an unmarried person.
One - the Word
One has possibly the most unusual English pronunciation of any word. Not only does it appear to begin with a 'W', but it sounds, to Southerners at least, like it should have a 'U' in the middle. The word itself can be traced back to an old Germanic language, but its root is a lot older than that. It is interesting that in most European languages the word for one derives from this root, for example the French un, the German ein and the Dutch een, as well as the Latin unus and the Greek oinos. Before the 14th Century, the English pronounced it like the first part of 'only', with some dialects saying 'un'. From that time, the 'wun' pronunciation started to spread across the land from the south-west, and it became almost universal in the 18th Century.
As well as a simple quantity, the word is used as the indefinite pronoun, in the same way as the French on or German man, but this is seen as somewhat elitist in colloquial English today. One wouldn't generally use it, would one?
We've mentioned many of the one-related words deriving from the Latin unus: unity, union, unique. Even the onion derives from this root, through the sense of it being a single, layered vegetable rather than having multiple cloves, as in garlic. There is an alternative set of one-words deriving from the Greek mono: monochrome, monotone, monogamous, etc.
On the printed page, the 'one' symbol often looks like an uppercase letter 'I', or a lowercase 'l', and, depending on your font, you may not even be able to distinguish these, the ninth and twelfth letters of the alphabet. Many early typewriters had no separate key for the figure one.
You only live once, unless you're James Bond, or believe in reincarnation, and there are a number of very singular life events. We have one mother and one father. We are born once, die once, and have one funeral (with the odd notable exception).
What are the units of life? We have one soul or spirit, and many of us believe in one creator being. A biologist may see the unit of life as being the cell, and indeed we have identified single-cellular organisms through the microscope. At the larger scale, we humans have an external duplex symmetry to our bodies, but many of our internal organs are singular: our heart, liver and brain, for example.
A physicist or chemist may see the fundamental unit as the molecule, or the atom, or indeed some unreachable sub-atomic particle.
One or Two Items of Trivia
Unity was once a popular girl's name, particularly among Puritans, signifying the quality of being one. It's also used as an anglicisation of the Irish name Úna, who according to legend was the mother of the hero Conn Céatchathach (Conn of the Hundred Battles). Úna has also been anglicised as Juno, Agnes and even Winifred.
A number of hit songs have been written with the title 'One', including versions recorded by Metallica and U2, but none has yet reached Number One in the UK Hit Parade.
A number of alternative and slang terms for one have grown up around sports and games. A onesie is a move in jacks, in which one jack is caught (it's often followed by a twosie, threesie, etc). A oner is a conker which has beaten one other zero-rated conker. We also have the ace - the playing card or the side of the dice with only one spot. In cricket, one run is known as a single, and in cards, one of a suit is a singleton.
One point is sometimes all you need to win a sporting contest, and One-Upmanship is one of a series of satirical self-help books by Stephen Potter, which were made into a 1970s BBC comedy series, starring Richard Briers and Peter Jones1.