The Math Book

by DK

  While there have been many different types of musical scales in use by different cultures, the long tradition of Western music dates back to the Pythagoreans and their quest to understand the relationship between music and mathematical proportions.

  The numerology of the Divine Comedy by Dante (1265–1331)—pictured here in a fresco from the Duomo in Florence, Italy—reflects the influence of Pythagoras, whom Dante mentions several times in his writings.

  The legacy of Pythagoras

  Pythagoras’s status as the most famous mathematician from antiquity is justified by his contributions to geometry, number theory, and music. His ideas were not always original, but the rigor with which he and his followers developed them, using axioms and logic to build a system of mathematics, was a fine legacy for those who succeeded him.

  There is geometry in the humming of the strings, there is music in the spacing of the spheres.

  Pythagoras

  PYTHAGORAS

  Pythagoras was born around 570 BCE on the Greek island of Samos in the eastern Aegean Sea. His ideas have influenced many of the greatest scholars in history, from Plato to Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagoras is thought to have traveled widely, assimilating ideas from scholars in Egypt and elsewhere in the Middle East before establishing his community of around 600 people in Croton, southern Italy, around 518 BCE. This ascetic brotherhood required its members to live for intellectual pursuits, while following strict rules of diet and clothing. It is from this time onward that his theorem and other discoveries were probably set down, although no records remain. At the age of 60, Pythagoras is said to have married a young member of the community, Theano, and perhaps had two or three children. Political upheaval in Croton led to a revolt against the Pythagoreans. Pythagoras may have been killed when his school was set on fire, or shortly afterward. He is said to have died around 495 BCE.

  See also: Irrational numbers • The Platonic solids • Syllogistic logic • Calculating pi • Trigonometry • The golden ratio • Projective geometry

  IN CONTEXT

  KEY FIGURE

  Hippasus (5th century BCE)

  FIELD

  Number systems

  BEFORE

  19th century BCE Cuneiform inscriptions show that the Babylonians constructed right-angled triangles and understood their properties.

  6th century BCE In Greece, the relationship between the side lengths of a right-angled triangle is discovered, and is later attributed to Pythagoras.

  AFTER

  400 BCE Theodorus of Cyrene proves the irrationality of the square roots of the nonsquare numbers between 3 and 17.

  4th century BCE The Greek mathematician Eudoxus of Cnidus establishes a strong mathematical foundation for irrational numbers.

  Any number that can be expressed as a ratio of two integers—a fraction, a decimal that either ends or repeats in a recurring pattern, or a percentage—is said to be a rational number. All whole numbers are rational as they can be shown as fractions divided by 1. Irrational numbers, however, cannot be expressed as a ratio of two numbers

  Hippasus, a Greek scholar, is believed to have first identified irrational numbers in the 5th century BCE, as he worked on geometrical problems. He was familiar with Pythagoras’s theorem, which states that the square of the hypotenuse in a right-angled triangle is equal to the sum of the squares of the other two sides. He applied the theorem to a right-angled triangle that has both shorter sides equal to 1. As 12 + 12 = 2, the length of the hypotenuse is the square root of 2.

  Hippasus realized, however, that the square root of 2 could not be expressed as the ratio of two whole numbers—that is, it could not be written as a fraction, as there is no rational number that can be multiplied by itself to produce precisely 2. This makes the square root of 2 an irrational number, and 2 itself is termed nonsquare or square-free. The numbers 3, 5, 7, and many others are similarly nonsquare and in each case, their square root is irrational. By contrast, numbers such as 4 (22), 9 (32), and 16 (42) are square numbers, with square roots that are also whole numbers and therefore rational.

  The concept of irrational numbers was not readily accepted, although later Greek and Indian mathematicians explored their properties. In the 9th century, Arab scholars used them in algebra.

  Hippasus may have encountered irrational numbers while exploring the relationship between the length of the side of a pentagon and one side of a pentagram formed inside it. He found that it was impossible to express it as a ratio between two whole numbers.

  In decimal terms

  The positional decimal system of Hindu–Arabic numeration allowed further study of irrational numbers, which can be shown as an infinite series of digits after the decimal point with no recurring pattern. For example, 0.1010010001… with an extra zero between each successive pair of 1s, continuing indefinitely, is an irrational number. Pi (π), which is the ratio of the circumference of a circle to its diameter, is irrational. This was proved in 1761 by Johann Heinrich Lambert—earlier estimations of π had been 3 or 22⁄7.

  Between any two rational numbers, another rational number can always be found. The average of the two numbers will also be rational, as will the average of that number and either of the original numbers. Irrational numbers can also be found between any two rational numbers. One method is to change a digit in a recurring sequence. For example, an irrational number can be found between the recurring numbers 0.124124… and 0.125125… by changing 1 to 3 in the second cycle of 124, to give 0.124324…, and doing so again at the fifth, then ninth cycle, increasing the gap between the replacement 3s by one cycle each time.

  One of the great challenges of modern number theory has been establishing whether there are more rational or irrational numbers. Set theory strongly indicates that there are many more irrational numbers than rational numbers, even though there are infinite numbers of each.

  HIPPASUS

  Details of Hippasus’s early life are sketchy, but it is thought that he was born in Metapontum, in Magna Graecia (now southern Italy), around 500 BCE. According to the philosopher Iamblichus, who wrote a biography of Pythagoras, Hippasus was a founder of a Pythagorean sect called the Mathematici, which fervently believed that all numbers were rational.

  Hippasus is usually credited with discovering irrational numbers, an idea that would have been considered heresy by the sect. According to one story, Hippasus drowned when his fellow Pythagoreans threw him over the side of a boat in disgust. Another story suggests that a fellow Pythagorean discovered irrational numbers, but Hippasus was punished for telling the outside world about them. The year of Hippasus’s death is not known but is likely to have been in the 5th century BCE.

  Key work

  5th century BCE Mystic Discourse

  See also: Positional numbers • Quadratic equations • Pythagoras • Imaginary and complex numbers • Euler’s number

  IN CONTEXT

  KEY FIGURE

  Zeno of Elea (c. 495–430 BCE)

  FIELD

  Logic

  BEFORE

  Early 5th century BCE The Greek philosopher Parmenides founds the Eleatic school of philosophy in Elea, a Greek colony in southern Italy.

  AFTER

  350 BCE Aristotle produces his treatise Physics, in which he draws on the concept of relative motion to refute Zeno’s paradoxes.

  1914 British philosopher Bertrand Russell, who described Zeno’s paradoxes as immeasurably subtle, states that motion is a function of position with respect to time.

  Zeno of Elea belonged to the Eleatic school of philosophy that flourished in ancient Greece in the 5th century BCE. In contrast to the pluralists, who believed that the Universe could be divided into its constituent atoms, Eleatics believed in the indivisibility of all things.

  Zeno wrote 40 paradoxes to show the absurdity of the pluralist view. Four of these—the dichotomy paradox, Achilles and the tortoise, the arrow paradox, and the stadium paradox—address motion. The dichotomy paradox s
hows the absurdity of the pluralist view that motion can be divided. A body moving a certain distance, it says, would have to reach the halfway point before it arrived at the end, and in order to reach that halfway mark, it would first have to reach the quarter-way mark, and so on ad infinitum. Because the body has to pass through an infinite number of points, it would never reach its goal.

  In the paradox of Achilles and the tortoise, Achilles, who is 100 times faster than the tortoise, gives the creature a head start of 100 meters in a race. At the sound of the starting signal, Achilles runs 100 meters to reach the tortoise’s starting point, while the tortoise runs 1 meter, giving it a 1 meter lead. Undeterred, Achilles runs another meter; however, in the same time, the tortoise runs one-hundredth of a meter, so it is still in the lead. This continues, and Achilles never catches up.

  The stadium paradox concerns three columns of people, each containing an equal number of people; one group is at rest, while the other two run past each other at the same speed in opposite directions. According to the paradox, a person in one moving group can pass two people in the other moving group in a fixed time, but only one person in the stationary group. The paradoxical conclusion is that half a given time is equivalent to double that time.

  Over the centuries, many mathematicians have refuted the paradoxes. The development of calculus allowed mathematicians to deal with infinitesimal quantities without resulting in contradiction.

  The paradox of Achilles and the tortoise maintains that a fast object, such as Achilles, will never catch up with a slow one, such as a tortoise. Achilles will get closer to the tortoise, but never actually overtake it.

  ZENO OF ELEA

  Zeno of Elea was born around 495 BCE in the Greek city of Elea (now Velia, in southern Italy). At a young age, he was adopted by the philosopher Parmenides, and was said to have been “beloved” by him. Zeno was inducted into the school of Eleatic thought, founded by Parmenides. At the age of around 40, Zeno traveled to Athens, where he met Socrates. Zeno introduced the Socratic philosophers to Eleatic ideas.

  Zeno was renowned for his paradoxes, which contributed to the development of mathematical rigor. Aristotle later described him as the inventor of the dialectical method (a method starting from two opposing viewpoints) of logical argument. Zeno collected his arguments in a book, but this did not survive. The paradoxes are known from Aristotle’s treatise Physics, which lists nine of them.

  Although little is known of Zeno’s life, the ancient Greek biographer Diogenes claimed he was beaten to death for trying to overthrow the tyrant Nearchus. In a clash with Nearchus, Zeno is reported to have bitten off the man’s ear.

  See also: Pythagoras • Syllogistic logic • Calculus • Transfinite numbers • The logic of mathematics • The infinite monkey theorem

  IN CONTEXT

  KEY FIGURE

  Plato (c. 428–348 BCE)

  FIELD

  Geometry

  BEFORE

  6th century BCE Pythagoras identifies the tetrahedron, cube, and dodecahedron.

  4th century BCE Theaetetus, an Athenian contemporary of Plato, discusses the octahedron and icosahedron.

  AFTER

  c. 300 BCE Euclid’s Elements fully describes the five regular convex polyhedra.

  1596 German astronomer Johannes Kepler proposes a model of the Solar System, explaining it geometrically in terms of Platonic solids.

  1735 Leonhard Euler devises a formula that links the faces, vertices, and edges of polyhedra.

  The perfect symmetry of the five Platonic solids was probably known to scholars long before the Greek philosopher Plato popularized the forms in his dialogue Timaeus, written in c. 360 BCE. Each of the five regular convex polyhedra—3-D shapes with flat faces and straight edges—has its own set of identical polygonal faces, the same number of faces meeting at each vertex, as well as equilateral sides, and same-sized angles. Theorizing on the nature of the world, Plato assigned four of the shapes to the classical elements: the cube (also known as a regular hexahedron) was associated with earth; the icosahedron with water; the octahedron with air; and the tetrahedron with fire. The 12-faced dodecahedron was associated with the heavens and its constellations.

  Composed of polygons

  Only five regular polyhedra are possible—each one created either from identical equilateral triangles, squares, or regular pentagons, as Euclid explained in Book XIII of his Elements. To create a Platonic solid, a minimum of three identical polygons must meet at a vertex, so the simplest is a tetrahedron— a pyramid made up of four equilateral triangles. Octahedra and icosahedra are also formed with equilateral triangles, while cubes are created from squares, and dodecahedra are constructed with regular pentagons.

  Platonic solids also display duality: the vertices of one polyhedron correspond to the faces of another. For example, a cube, which has six faces and eight vertices, and an octahedron (eight faces and six vertices) form a dual pair. A dodecahedron (12 faces and 20 vertices), and an icosahedron (20 faces and 12 vertices) form another dual pair. Tetrahedra, which have four faces and four vertices, are said to be self-dual.

  Shapes in the Universe?

  Like Plato, later scholars sought Platonic solids in nature and the Universe. In 1596, Johannes Kepler reasoned that the positions of the six planets then known (Mercury, Venus, Earth, Mars, Jupiter, and Saturn) could be explained in terms of the Platonic solids. Kepler later acknowledged he was wrong, but his calculations led him to discover that planets have elliptical orbits.

  In 1735, Swiss mathematician Leonhard Euler noted a further property of Platonic solids, later shown to be true for all polyhedra. The sum of the vertices (V) minus the number of edges (E) plus the number of faces (F) always equals 2, that is, V ˗ E + F = 2.

  It is also now known that Platonic solids are indeed found in nature—in certain crystals, viruses, gases, and the clustering of galaxies.

  PLATO

  Born around 428 BCE to wealthy Athenian parents, Plato was a student of Socrates, who was also a family friend. Socrates’ execution in 399 BCE deeply affected Plato and he left Greece to travel. During this period his discovery of the work of Pythagoras inspired a love of mathematics. Returning to Athens, in 387 BCE he founded the Academy, inscribing over its entrance the words “Let no one ignorant of geometry enter here.” Teaching mathematics as a branch of philosophy, Plato emphasized the importance of geometry, believing that its forms—especially the five regular convex polyhedra—could explain the properties of the Universe. Plato found perfection in mathematical objects, believing they were the key to understanding the differences between the real and the abstract. He died in Athens around 348 BCE.

  Key works

  c. 375 BCE The Republic

  c. 360 BCE Philebus

  c. 360 BCE Timaeus

  See also: Pythagoras • Euclid’s Elements • Conic sections • Trigonometry • Non-Euclidean geometries • Topology • The Penrose tile

  IN CONTEXT

  KEY FIGURE

  Aristotle (384–322 BCE)

  FIELD

  Logic

  BEFORE

  6th century BCE Pythagoras and his followers develop a systematic method of proof for geometric theorems.

  AFTER

  c. 300 BCE Euclid’s Elements describes geometry in terms of logical deduction from axioms.

  1677 Gottfried Leibniz suggests a form of symbolic notation for logic, anticipating the development of mathematical logic.

  1854 George Boole publishes The Laws of Thought, his second book on algebraic logic.

  1884 The Foundations of Arithmetic by German mathematician Gottlob Frege examines the logical principles underpinning mathematics.

  In the Square of Opposition, S is a subject, such as “sugar,” and P a predicate, such as “sweet.” A and O are contradictory, as are E and I (if one is true, the other is false, and vice versa). A and E are contrary (both cannot be true but both can be false); I and O are subcontrary: both can be true but both cannot be false. I
is a subaltern of A and O is a subaltern of E. In syllogistic logic, this means that if A is true, I must be true, but that if I is false, A must be false as well.

  In Classical Greece, there was no clear distinction between mathematics and philosophy; the two were considered interdependent. For philosophers, one important principle was the formulation of cogent arguments that followed a logical progression of ideas. The principle was based on Socrates’ dialectal method of questioning assumptions to expose inconsistencies and contradictions. Aristotle, however, did not find this model entirely satisfactory, so he set about determining a systematic structure for logical argument. First, he identified the different kinds of proposition that can be used in logical arguments, and how they can be combined to reach a logical conclusion. In Prior Analytics, he describes the propositions as being of broadly four types, in the form of “all S are P,” “no S are P,” “some S are P,” and “some S are not P,” where S is a subject, such as sugar, and P the predicate—a quality, such as sweet. From just two such propositions an argument can be constructed and a conclusion deduced. This is, in essence, the logical form known as the syllogism: two premises leading to a conclusion. Aristotle identified the structure of syllogisms that are logically valid, those where the conclusion follows from the premises, and those that are not, where the conclusion does not follow from the premises, providing a method for both constructing and analyzing logical arguments.