From 0 to Infinity in 26 Centuries (5 page)
Read From 0 to Infinity in 26 Centuries Online
Authors: Chris Waring
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:
h
2
= a
2
+ b
2
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 = x
2
/y
2
. This means that x
2
must be twice y
2
and so x
2
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 x
2
must be twice y
2
, then 4w
2
= 2y
2
, so 2w
2
= y
2
, and so y
2
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.
S
OCRATES
(
c.
470–399
BC
), P
LATO
(427–347
BC
)
AND
A
RISTOTLE
(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.
Another 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.