From 0 to Infinity in 26 Centuries Read Online Free PDF Page A

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 toset 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 otherPlatonic 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