From 0 to Infinity in 26 Centuries

by Chris Waring

  When it came to mathematics, Thales was, like many Ancient Greeks, interested in geometry. He understood the principle of similar triangles and used it to predict the height of the pyramids.

  The two triangles on the preceding page are similar triangles because their angles are the same, and therefore their sides must be in proportion to each other. Thales was able to determine the height of a pyramid by measuring the length of its shadow. He waited until his own shadow was the same length as his height to measure the pyramid’s shadow in order to determine how tall it was.

  PYTHAGORAS (c. 570–c. 490 BC)

  Pythagoras’ name is instantly recognizable because his theorem has been taught to students of mathematics the world over. Because he left no written works behind, everything we know about Pythagoras was written long after he died. He founded a religious movement called Pythagoreanism, hence much that was written about him, embellished over time, assumed a decidedly mystical tinge. Among other things, Pythagoras is described as having a thigh made of gold and the ability to be in two places at once.

  The theorem

  Today Pythagoras is best known for his theorem for working out the hypotenuse (the longest side, the side opposite the right angle) of a right-angled triangle. In words, Pythagoras’ theorem is described as:

  The square on the hypotenuse is equal to the sum of the squares on the other two sides.

  But it’s much pithier to say:

  h2 = a2 + b2

  Pythagoras wasn’t the first to explore the theorem. The Ancient Egyptians had investigated this idea, and the Mesopotamians too – some of their homework tablets were Pythagoras-type questions (see here). We know from several different sources that, as a young man, Pythagoras travelled extensively around the Mediterranean, and possibly even further afield, to gather knowledge, so it seems he had plenty of opportunity to glean this information from elsewhere.

  So why is the theorem named after him? Because Pythagoras was the first Greek identified with the concept. And, since we were unable to read hieroglyphics or cuneiform until recently, it was assumed Pythagoras must have worked the theorem out independently.

  Many Strings to His Bow

  Another legend associated with Pythagoras is that he was the first person to work out the relationship between a length of string and the note it produces when plucked. He also noted that if the lengths of two strings were in a whole number ratio to each other then a harmonious chord was produced.

  Pythagoras is also often attributed to the discovery of the Platonic solids and the golden mean, both of which were recorded by Plato (see here).

  Pythagoreanism

  Pythagoras’ religious cult was a somewhat bizarre group. Its members favoured an ascetic lifestyle by avoiding talking and following a vegetarian diet. They were a highly secretive group – the revelation of a cult secret was punishable by death. Pythagoreans were also very exclusive, and managed to antagonize the inhabitants of nearby towns enough for the populace to burn down their meeting place, killing many of the cult’s members in the process.

  Central to the Pythagoreans’ doctrine was the idea that numbers were divine. They also believed that all numbers could be written as fractions. One notable legend centres on an unfortunate fellow called Hippasus, a Pythagorean who was pretty certain he had come across numbers that could not be written as fractions. In some of the more fanciful legends, Pythagoras asks Hippasus to take a boat out to sea with him to discuss his heretical ideas – only Pythagoras comes back. An alternative version sees Hippasus being drowned by the gods for his crimes against holy numbers.

  An irrational discovery

  Whatever the circumstances of Hippasus’ sticky end, he may have been one of the first people to discover irrational numbers – numbers that cannot be written as a fraction, and whose decimal equivalent goes on for ever without repeating.

  See if you can follow some good ol’ Greek reductio ad absurdum (see here) to see why √2 must be an irrational number.

  √2 is the square root of 2 – the number that when multiplied by itself gives an answer of 2.

  √2 = 1.4142135623...

  If √2 can be written as a fraction, let’s say it is x/y in its lowest terms.

  If x/y is in its lowest terms x and y cannot both be even because if they were even you would be able to divide both x and y by 2, so they could not have been in their lowest terms to start with.

  If you square everything you get 2 = x2/y2. This means that x2 must be twice y2 and so x2 must be even because it is two times something. This in turn means x must be even because odd x odd = odd.

  If x is even then y must be odd because, as you might recall, x/y was in its lowest terms and the two, therefore, cannot both be even.

  If x is even, then it must be divisible by 2. So let’s say x = 2 × w.

  If x = 2 × w and x2 must be twice y2, then 4w2 = 2y2, so 2w2 = y2, and so y2 must be even because it is twice something. It follows that y must be even, which conflicts with our earlier deduction that y must be odd!

  If x is even, then y must also be even. But we said it must be odd. So √2 ≠ x/y so √2 cannot be written as a fraction.

  SOCRATES (c. 470–399 BC), PLATO (427–347 BC) AND ARISTOTLE (384–322 BC)

  Three of Ancient Greece’s most renowned philosophers, Socrates, Plato and Aristotle are often mentioned together because Socrates taught Plato, who in turn taught Aristotle. They were hugely influential to Western thought because, essentially, they were responsible for inventing it.

  Socrates

  While he did not contribute to mathematics directly, Socrates did supply a way of thinking about problems, called the Socratic method, which provided a logical framework for solving mathematical conundrums. Using the Socratic method, a difficult problem could be broken down into a series of smaller, more manageable pieces; by working through these smaller challenges the inquirer would eventually reach a solution to the main problem. Although Socrates generally used this method to solve ethical questions, it is equally useful for mathematical and scientific problems.

  Plato’s dialogues

  Plato was a student of Socrates’, and is well known for writing a series of works called the Socratic Dialogues, which use a fictional discussion between Socrates and a range of other people to set forth ideas and philosophies; a little like reading a fictional transcript of a lesson, with a student questioning the ideas put forth by his teacher.

  In one such dialogue, Timaeus, written in c. 360 BC, Plato discusses several important mathematic and scientific ideas.

  Plato’s solids

  The elements is the first topic addressed in Timaeus. Today modern atomic theory tells us that there are over 100 elements that can be combined to create all known substances. In his dialogue Plato was the first to propose that the four elements – fire, air, water and earth – each assume a specific shape. We name the shapes of these elements the Platonic solids in his honour.

  The Platonic solids are 3D shapes (polyhedrons) whose faces are made up of regular (all sides and angles are equal) 2D shapes (polygons). For example, a triangle-based pyramid made up of equilateral triangles is a Platonic solid called a tetrahedron.

  These four elements, much like the elements we know today, could be combined to make any substance. There is one other Platonic solid – the twelve-sided dodecahedron– which was not an element, but which represented the shape of the universe.

  Going for gold

  Another important concept discussed in Timaeus is the golden mean, sometimes called the golden ratio or golden section.

  The golden mean is the optimum position between two extremes, and it’s also a number: 1.6180339887... – one of those irrational numbers the Pythagoreans were not very keen on. Much like √2, the golden mean cannot be written as a fraction because its decimal continues for ever without repeating. This is inconvenient when it comes to writing the number down, so mathematicians use the symbol φ (the Greek letter phi) to represent it.

  Ano
ther irrational number that has its own Greek letter is 3.14159265...: π, which you will remember from learning about circles at school. We get π by dividing a circle’s circumference (its perimeter) by its diameter (the distance across the circle through the centre). It doesn’t matter what the size of the circle is, you always get the same value: π. φ has a similar geometrical provenance.

  If a line is divided into a longer part and a shorter part, and if the total length (x + y) divided by x gives the same value as x ÷ y then the line has been split into the golden ratio. As with π, the length of the line doesn’t matter – to get it to work you find that x ÷ y = 1.618... = φ.

  The same idea works with shapes too. The Ancient Greeks considered a rectangle with its longer side φ times longer than its shorter side to be the most aesthetically pleasing rectangle possible.

  It is often said that many important examples of sculpture and architecture are made using the golden mean.

  The Parthenon was designed according to the golden mean. Its length and height and the space between the columns were designed in perfect proportion to one other.

  Aristotle

  The son of the doctor at the court of the kings of Macedon, Aristotle was a nobleman who became a hugely influential philosopher. He was taught by Plato and later became a teacher at Plato’s Academy, and he contributed ideas on a whole host of subjects, from politics and ethics to physics and zoology. So wide-ranging were his skills, it has been suggested Aristotle knew everything it was possible to know. Indeed, his influence extended through to the philosophy of the modern world.

  Zero Option

  Aristotle let things slip in his treatment of numbers. He felt that a number only really had meaning if it was an amount of something: a pile. In Aristotle’s eyes, 10 apples, 1 apple, ½ an apple and 1/10 of an apple were all valid numbers. However, if you do not have an apple, you have nothing to pile up or count – zero, as far as Aristotle was concerned, was not a number.

  Aristotle is known chiefly for his logic, a series of works that comprised the earliest-known study of the theory of logic. His theories have since split into many different branches, some highly mathematical, others more philosophical.

  Aristotle’s work in mathematics and science focused on explaining the way things behave by describing them rather than using numbers and equations. He was among the first to explain the motion of objects (a subject we today call kinetics, from the Greek for ‘movement’). Aristotle’s descriptions acknowledged that time and space are not arranged in indivisible chunks but are continuous, which allowed him to show that Zeno of Elea’s ideas were flawed and that Achilles would have been able to catch up with the tortoise!

  EUCLID (c. 325–c. 265 BC)

  While little is known about the Greek mathematician Euclid, we do know that he was active in Alexandria in Egypt under Greek rule, and is notable for having penned a groundbreaking book called Elements. Certainly one of the most important maths books of all time, Euclid’s Elements was considered essential reading for any scholar well into the nineteenth century.

  Elementary proof

  Although Euclid drew on the ideas of others, he was one of the first mathematicians to produce work that used mathematical logic in order to prove theories. This idea of proof is one of the foundations of mathematics.

  Elements covers much of geometry and ideas about numbers, including prime numbers and other number sequences, and all of Euclid’s geometrical constructions were made using only a pair of compasses and a straight edge.

  It is split into thirteen books, each of which starts with definitions of words to help make it clear what Euclid means when he refers to words such as point, line, straight, surface, etc. Euclid then sets out a list of axioms or statements that are evidently true, such as ‘all right angles are equal to each other’ and ‘if A=B and A=C, then B=C’.

  The next section of the Elements is called ‘Propositions’, in which Euclid proposes a method of how to carry out a mathematical task. For example, in Proposition 1 of Book 1 Euclid shows how to draw an equilateral triangle (all the sides are the same length and all the angles are equal to 60°), and he then goes on to prove that the triangle is, in fact, equilateral.

  ERATOSTHENES (276–195 BC)

  It would be wrong to talk too much about prime numbers without mentioning multi-disciplined mathematician Eratosthenes, who hailed from a Greek city in modern-day Libya. He was responsible for many great intellectual endeavours, including calculating the earth’s circumference to a surprising degree of accuracy and coining the word ‘geography’, which means ‘drawing the earth’ in Ancient Greek. Mathematically, Eratosthenes’ greatest contribution is the Sieve of Eratosthenes.

  In their prime

  Before we look at the sieve let us first contemplate prime numbers: numbers that have only two factors – themselves and 1. Hence 13 is a prime number because 1 and 13 are the only numbers that divide into it without leaving a remainder. 9 is not prime, because it can be divided by 1, 3 and 9, which means it has three factors. 1 is also not a prime number because it has only one factor.

  Prime numbers are important for two reasons:

  1. Any whole number or integer greater than 1 can be written as a chain of multiplied prime numbers. For example, the numbers between 20 and 30 can be written as follows:

  20 = 2 × 2 × 5

  21 = 3 × 7

  22 = 2 × 11

  23 = 23 (prime)

  24 = 2 × 2 × 2 × 3

  25 = 5 × 5

  26 = 2 × 13

  27 = 3 × 3 × 3

  28 = 2 × 2 × 7

  29 = 29 (prime)

  30 = 2 × 3 × 5

  There is only one way of doing this for each number so it seems to me, at least, that primes are the equivalent of DNA for numbers.

  Fundamentals

  The idea that any whole number greater than 1 can be expressed as the unique product of a chain of multiplied prime numbers is called the fundamental theorem of arithmetic.

  2. They are very mysterious – there is no pattern to prime numbers, and there is no formula that will produce them. To this day the nature of prime numbers is still under intensive study by mathematicians.

  Eratosthenes’ sieve works using a very simple principle to help find prime numbers up to a certain limit. 2 is the first prime number. Anything that 2 goes into cannot be prime, because it would then have 2 as a factor as well as itself and 1.

  If we set ourselves a limit of 100, we could highlight 2 as a prime and then eliminate all the numbers that have 2 as a factor: 4, 6, 8, etc. up to 100. If we use a grid we can shade them in to generate a pattern:

  You don’t even need to be brilliant at your two-times table to do this – you could just count on 2 each time and shade in each square you land on.

  After you’ve shaded in all the multiples of 2 you move on to the next unshaded number, which also happens to be the next prime number: 3. We highlight 3 as a prime and then eliminate all the multiples of 3, some of which have already been eliminated in the first round. The next unshaded number is 5, which again is also prime. As before, highlight that and then eliminate the multiples of 5.

  As you move along, the next unshaded number must be prime because none of the prime numbers that went before it could go into it. If you keep on repeating this process eventually you’ll have a completed sieve. Turn to page 55 to see what this looks like.

  To Infinity and Beyond

  Euclid’s theorem demonstrated that there are infinitely many prime numbers. We know that any number can be made by multiplying a chain of prime numbers together; thanks to our sieve, we also now know all of the prime numbers under 100. How can we be sure there are more? Let’s use the Sieve of Eratosthenes to investigate.

  If you multiply all the primes together you generate a number. This next number will either be a prime number or it won’t. If the next number in sequence is a prime then we have a new prime number.

  However, if the next number isn’t prime there
must be a prime number that we don’t already know of that goes into making it – therefore there’s another prime number somewhere.

  So, whatever happens, we either have a new prime number or know there is an unknown prime number that is less than or more than our number. No matter how large we make the sieve, there is always another prime number that is not on it, therefore there must be an infinite number of primes.

  You can make the sieve as big as you like in order to work out higher and higher prime numbers. There are no hard calculations to do, but it is quite a tedious process – something a Greek mathematician would probably have left to an educated slave.

  ARCHIMEDES (287–212 BC)

  Archimedes was a friend of Eratosthenes and he hailed from the city of Syracuse in present-day Sicily. He was famous as a scientist and engineer: he invented the Archimedes screw for pumping liquids and raising grain, which is still in use today. Archimedes is also said to have defended Syracuse from Roman warships by directing an intense ray of light from the sun towards the approaching soldiers, setting their vessels alight.

  From straight to circular

  Archimedes’ contributions to mathematics are no less impressive, even if they are less well known. He worked out a value for π by noting that, as a polygon accrues more sides, it gets closer and closer to becoming a circle. π is defined as a circle’s circumference divided by its diameter. It is hard to measure the curved edge of a circle accurately, but easy to measure the straight sides of a polygon to find the perimeter. By approximating a circle as a polygon with a certain number of sides, Archimedes was able to find a value for π by dividing the polygon’s perimeter by the distance across the polygon. Archimedes performed this calculation with a polygon that had up to 96 sides. During his investigations he came up with a value of between 3.143 and 3.140 for π, which is pretty close to its actual value: 3.1415...