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3. Everything / Maths, Science & Technology / Mathematics

Two - the Dichotomous Number

Twin boys.

Two is an odd number. Well, it's not an odd number in that sense. It's definitely even; in fact it's the first even number which we can count (we don't count zero – it's not a natural or 'counting' number). It's even because it divides by two – but then it is two! Now, we're not saying that it's even because it divides by itself – that's certainly not correct, otherwise three would also be even. No it's even because we say that all the even numbers divide by two. So that's why it's odd. No, not odd, rather 'special'.

You see, we just like dividing things into two – it's one of those little human traits we have: categorising things. Of course we sometimes categorise things in more than two ways, but we just naturally prefer putting them in one of two boxes. Something is up or down. Or maybe it's left or right. Or is it good or evil? Sometimes there's just no other box to put things in.

This categorisation into one of two sets is known as dichotomy. Humans are naturally dichotomous in a number of ways – our bodies are laterally symmetrical, so we have two eyes, two arms, two legs, etc, and at times we need to differentiate one from the other, which is where left and right come in. Maybe if we had three of everything we would be trichotomous and would need a third state, but the point is: we're not, so we don't tend to.

Dichotomy is the principle behind the biological key classification, used to identify specimens by means of a series of questions, each having only two answers. This method is attributed to the Greek philosopher and scientist Aristotle (384 - 322 BC).

Putting Two and Two Together

The Ancient Greeks were undoubtedly skilled mathematicians. They were also very philosophical, but somewhat sceptical about the number two. They didn't have a number one as such – that was the basic fundamental unit of which other numbers were made – but two was also different from the other numbers. You see, the Greeks were very visual people, and considered numbers geometrically; they were lengths of lines, or points to be arranged in space. When they arranged numbers as points in a straight line, this line would start with a point, end with a point and have some points in the middle. But not two – it started, and it ended, but it just didn't seem to contain anything, and so the Greeks were somewhat unsure whether it was indeed a number.

There was one further reason to be doubtful of two. The Greeks were very interested in the basic operations of mathematics, namely addition and multiplication, and the latter was seen to be a more powerful operation as it resulted in bigger numbers. Two, however, didn't play along: when you multiplied by two, you simply got the same number as if you added the number to itself. Now, you might think this is a fairly odd way of looking at things – surely that's the definition of multiplying by two – but to the Greeks, this was just another anomaly which marred the perfection of their beautiful mathematical world.

Some Mathematical Properties

Perhaps the Greeks were somewhat missing the point with operations. One reason two is rather special is because the basic mathematical operations work on two numbers. We add or multiply two numbers together, or we subtract one from another, or divide one into another. We call them binary operators for this reason – they consist of two arguments. We also use the word binary, or base 2, to describe a complete number system which has only two symbols. Every number is rendered in 1s and 0s, rather than the ten symbols we are more familiar with in our decimal system. Two itself is rendered as '10', leading to the old joke:

There are only 10 types of people in the world: those who understand binary, and those who don't.

There are some other properties of two which mathematical number theorists might find slightly more interesting than the rest of us. Two is the first prime number, and it is also a factor of ten, which makes it simple to identify which numbers divide by two – we only need to check that the last digit divides by two.

Two is the only number for which the equation in Fermat's Last Theorem holds1. In other words, for xn + yn = zn, there are only solutions for n = 2, as we see in Pythagorean right-angled triangles, like 32 + 42 = 52. There are no solutions for higher powers of n.

We are very familiar with the powers of two: 2, 4, 8, 16, 32, 64, etc, and examination of a binary number shows us that we can express any other number as a sum of powers of two2. There is another unproven theorem, known as Goldbach's conjecture, that every even number can be expressed as the sum of two prime numbers.

Finally, the mathematicians Descartes and Euler, working independently, showed us that for most solid figures3, if you add the number of vertices (corners) to the number of faces (sides), then subtract the number of edges, you will get the answer two!

The Language of Two

One further dichotomy of the human race, not to mention many other forms of life, is our gender: male or female. This is how many languages categorise words, and it's also how the Greeks saw the numbers: even numbers were female and odd numbers male.

Two is the first number which you might describe as 'plural' rather than singular, and we recognise this in language by having a different case for plural words. However, in some early languages, notably Egyptian, Arabic, Hebrew, Sanskrit and Greek, there was singular case, then a separate case for exactly two of something, and then a further case for more than two.

Like 'one', two has a rather unusual English pronunciation. The Oxford English Dictionary traces its journey from the Old English version of the word, twá4: we'd have spoken it as 'twa', then 'tw', 'two', 'twu', then finally 'tu'. The older forms still appear in dialects of English, notably in Scotland and the North of England.

We have a number of English words which imply two or twoness:

  • Two derives from an old Germanic word, which itself comes from an older root which supplied many of the ancient and modern European forms, such as the French deux, the German zwei, the Dutch twee, the Latin duo and the Greek duo.

  • Dual, double and duplex plainly come from the Latin duo, and the prefixes bi- and di- derive from a similar root5.

  • Twin derives from the Germanic root, as do words like between and even twig, due to the sense of it being a forked stick.

  • Pair can be traced back to the Latin paria (equals), as it implies two items which are equally matched.

  • Brace in terms of, say, a brace of pheasants is less clear. It certainly comes from the old French word brace meaning 'arms', but the word then goes on a bit of a journey. It could be that the word 'embrace' gave us the verb 'to brace' implying to tighten something, then this was applied to things that tighten, like straps. It is also known that a pair of hunting dogs on leashes was known as a brace, and this may have then been applied to two animals of other kinds.

  • Couple comes via the Latin copula (a tie or connection), and was first applied to a married couple before it became a general synonym for two of something.

  • A tandem, the bicycle made for two, is itself a Latin word meaning 'at length', suggesting in a sense one behind the other. It was originally applied to a carriage pulled by two horses in that configuration.

  • The English ordinal version of two, second, is unusual in that it's not connected with the word 'two' as it is in other languages, like the French deuxieme and the German zweite. There was no Old English word for 'twoth' at all; instead they used 'other', but as this was ambiguous adopted the word 'second' instead. This word came via French from the Latin secundus, meaning 'following'.

Special Meanings

Two has a number of meanings to us, which we can explore through our use of phrases and proverbs:

We often use two when we have an alternative and a choice to make: we might be in two minds, then select the lesser of two evils, or we could deny a choice by saying there's no two ways about it.

Someone who cheats could be two-timing us, and we might describe a liar as two-faced. We might issue them a challenge by saying 'two can play at that game'. On the other hand, we might emphasise agreement by saying 'that makes two of us'.

Two can also be a helpful addition: we may have two bites of the cherry, have two strings to our bow or we may say two heads are better than one.

Finally, it can be used to emphasise a very small quantity. We might offer our two cents' worth or tuppenceworth, or put two and two together to draw a conclusion from simple facts.


1 Well, it holds for n = 1, but that's trivial.
2 To make odd numbers, we also need 1 in this list, and this is two raised to the power of zero.
3 They must have no holes or dents in them, and each face must be a polygon – a flat surface bounded by straight lines.
4 Twá was the feminine version of the word. The neuter was and the masculine was twéen, which survives today as the word 'twain' in old phrases like 'never the twain shall meet'.
5 Examples include 'dilemma' – a choice between two unfavourable alternatives.

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Entry Data
Entry ID: A45901749 (Edited)

Written and Researched by:
Icy North

Edited by:
U284


Date: 20   January   2009


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Referenced Guide Entries
Numbers
Mathematics
Evil from a Western Perspective
Binary Star Systems
Integers and Natural Numbers
Fermat's Last Theorem
The Oxford English Dictionary
Taxonomy
How to Be a Philosopher
Aristotle
Dichotomous Lives of Abuse Survivors
The Four Main Reasons Why Scientists are Sometimes Incomprehensible
Binary Digits
An Holistic View of Cherries
Common Pheasant and Relatives
Zero
A Brief History of Greece
Divisibility
Prime Numbers
Leonhard Euler - Mathematician
Lead Training Your Dog
Unity - the Number One


Related BBC Pages
BBC Two


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