From 0 to Infinity in 26 Centuries Read Online Free PDF

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.
    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 thanusing 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!
    E UCLID (
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