Alex's Adventures in Numberland - Alex Bellos [177]
Platonic solid: the five solids whose sides are all identical regular polygons; in other words, the tetrahedron, cube, octahedron, icosahedron and dodecahedron.
Polygon: a two-dimensional closed shape made up of a finite number of straight lines.
Postulate: a statement that is assumed to be true, and used as an axiom.
Power: an operation that determines how many times a number is to be multiplied by itself, so if 10 is to be multiplied by itself four times, one writes 104 and calls this ‘ten to the power of four’. Powers are not always natural numbers, but when one talks about the ‘powers of x’ it is assumed that one is referring only to the powers of x that are.
Prime number: a natural number that has only two divisors, itself and 1.
Probability: the chance of an event taking place, expressed as a fraction between 0 and 1.
Quadratic equation: an equation of the form ax2 + bx + c = 0, where a, b, and c are constants and a is non-zero.
Radius: a straight line from the centre of a circle to the circumference.
Randowalk: a visual interpretation of randomness, in which each random event is expressed as movement in a random direction.
Rational number: a number that can be expressed as a fraction.
Regression to the mean: the phenomenon that after extreme events less extreme ones are more likely.
Regular polygon: a polygon with sides of equal length and with equal internal angles.
Sequence: a list of numbers.
Series: the sum of the terms in a sequence.
Set: a collection of things.
Tessellation: an arrangement of tiles that fills a two-dimensional space completely with no overlaps.
Theorem: a statement that can be proved from other theorems and/or axioms.
Transcendental number: a number that cannot be expressed as a solution to a finite equation.
Uncountable infinity: an infinite set whose members cannot be put in a one-to-one correspondence with the natural numbers.
Unique solution: the situation when there is one and only one possible answer.
Variable: a quantity that can vary in value.
Vertex: where two lines meet to form an angle, or the angular points in a three-dimensional shape.
Appendix One
In order to see how Annairizi’s tiled squares prove Pythagoras’s Theorem, look at the marked triangle chapter 2. All that we need to do is to rearrange the square of the hy
potenuse into precisely the squares of the other two sides. The square of the hypotenuse is made up of five sections; three are light grey and two are dark grey. We can see by considering how the pattern repeats that the light grey sections make up exactly the square of one of the sides of the triangle, and that the dark grey sections make up the square of the other side.
For Leonardo’s proof, first we need to show that the shaded sections in (i) and (ii) below are equal. We do this by rotating the section around point P. The two sections have >Iname side lengths and angles and therefore must be the same. Then we need to show that this section is equal to the section in (iii). This must be true since it is made up of identical parts.
With this information, we can complete the proof. The reflection of the first shaded section and its mirror image across the dotted line consists of two identical right-angled triangles and the squares on its two smaller sides. This area must be equal to the area covered by the sections shaded in (ii) and (iii) together, which consists of two identical right-angled triangles and the square of the hypotenuse. If we subtract the area of two triangles from both of these cases, the square of the hypotenuse must be equal to the square on the other two sides.
Appendix Two
In a unit square, the diagonal has length . To show that this is irrational I will use a proof by contradiction in which it is assumed that is rational, and then I will show how this leads to a contradiction. If it is contradictory to say that is rational,