The Math Book

by DK

  Geometry is knowledge of the eternally existent.

  Pythagoras

  Ancient Greek mathematician

  Geometry and calculus

  A third major field of mathematics, geometry, is concerned with the concept of space, and the relationships of objects in space: the study of the shape, size, and position of figures. It evolved from the very practical business of describing the physical dimensions of things, in engineering and construction projects, measuring and apportioning plots of land, and astronomical observations for navigation and compiling calendars. A particular branch of geometry, trigonometry (the study of the properties of triangles), proved to be especially useful in these pursuits. Perhaps because of its very concrete nature, for many ancient civilizations, geometry was the cornerstone of mathematics, and provided a means of problem-solving and proof in other fields.

  This was particularly true of ancient Greece, where geometry and mathematics were virtually synonymous. The legacy of great mathematical philosophers such as Pythagoras, Plato, and Aristotle was consolidated by Euclid, whose principles of mathematics based on a combination of geometry and logic were accepted as the subject’s foundation for some 2,000 years. In the 1800s, however, alternatives to classical Euclidean geometry were proposed, opening up new areas of study, including topology, which examines the nature and properties not only of objects in space, but of space itself.

  Since the Classical period, mathematics had been concerned with static situations, or how things are at any given moment. It failed to offer a means of measuring or calculating continuous change. Calculus, developed independently by Gottfried Leibniz and Isaac Newton in the 1600s, provided an answer to this problem. The two branches of calculus, integral and differential, offered a method of analyzing such things as the slope of curves on a graph and the area beneath them as a way of describing and calculating change.

  The discovery of calculus opened up a field of analysis that later became particularly relevant to, for example, the theories of quantum mechanics and chaos theory in the 1900s.

  Revisiting logic

  The late 19th and early 20th centuries saw the emergence of another field of mathematics—the foundations of mathematics. This revived the link between philosophy and mathematics. Just as Euclid had done in the 3rd century BCE, scholars including Gottlob Frege and Bertrand Russell sought to discover the logical foundations on which mathematical principles are based. Their work inspired a re-examination of the nature of mathematics itself, how it works, and what its limits are. This study of basic mathematical concepts is perhaps the most abstract field, a sort of meta-mathematics, yet an essential adjunct to every other field of modern mathematics.

  In mathematics, the art of asking questions is more valuable than solving problems.

  Georg Cantor

  German mathematician

  New technology, new ideas

  The various fields of mathematics—arithmetic, algebra, geometry, calculus, and foundations—are worthy of study for their own sake, and the popular image of academic mathematics is that of an almost incomprehensible abstraction. But applications for mathematical discoveries have usually been found, and advances in science and technology have driven innovations in mathematical thinking.

  A prime example is the symbiotic relationship between mathematics and computers. Originally developed as a mechanical means of doing the “donkey work” of calculation to provide tables for mathematicians, astronomers and so on, the actual construction of computers required new mathematical thinking. It was mathematicians, as much as engineers, who provided the means of building mechanical, and then electronic computing devices, which in turn could be used as tools in the discovery of new mathematical ideas. No doubt, new applications for mathematical theorems will be found in the future too—and with numerous problems still unsolved, it seems that there is no end to the mathematical discoveries to be made.

  The story of mathematics is one of exploration of these different fields, and the discovery of new ones. But it is also the story of the explorers, the mathematicians who set out with a definite aim in mind, to find answers to unsolved problems, or to travel into unknown territory in search of new ideas—and those who simply stumbled upon an idea in the course of their mathematical journey, and were inspired to see where it would lead. Sometimes the discovery would come as a game-changing revelation, providing a way into unexplored fields; at other times it was a case of “standing on the shoulders of giants,” developing the ideas of previous thinkers, or finding practical applications for them.

  This book presents many of the “big ideas” in mathematics, from the earliest discoveries to the present day, explaining them in layperson’s language, where they came from, who discovered them, and what makes them significant. Some may be familiar, others less so. With an understanding of these ideas, and an insight into the people and societies in which they were discovered, we can gain an appreciation of not only the ubiquity and usefulness of mathematics, but also the elegance and beauty that mathematicians find in the subject.

  Mathematics, rightly viewed, possesses not only truth, but supreme beauty.

  Bertrand Russell

  British philosopher and mathematician

  INTRODUCTION

  As early as 40,000 years ago, humans were making tally marks on wood and bone as a means of counting. They undoubtedly had a rudimentary sense of number and arithmetic, but the history of mathematics only properly began with the development of numerical systems in early civilizations. The first of these emerged in the sixth millennium BCE, in Mesopotamia, western Asia, home to the world’s earliest agriculture and cities. Here, the Sumerians elaborated on the concept of tally marks, using different symbols to denote different quantities, which the Babylonians then developed into a sophisticated numerical system of cuneiform (wedge-shaped) characters. From about 4000 BCE, the Babylonians used elementary geometry and algebra to solve practical problems—such as building, engineering, and calculating land divisions—alongside the arithmetical skills they used to conduct commerce and levy taxes.

  A similar story emerges in the slightly later civilization of the ancient Egyptians. Their trade and taxation required a sophisticated numerical system, and their building and engineering works relied on both a means of measurement and some knowledge of geometry and algebra. The Egyptians were also able to use their mathematical skills in conjunction with observations of the heavens to calculate and predict astronomical and seasonal cycles and construct calendars for the religious and agricultural year. They established the study of the principles of arithmetic and geometry as early as 2000 BCE.

  Greek rigor

  The 6th century BCE onward saw a rapid rise in the influence of ancient Greece across the eastern Mediterranean. Greek scholars quickly assimilated the mathematical ideas of the Babylonians and Egyptians. The Greeks used a numerical system of base-10 (with ten symbols) derived from the Egyptians. Geometry in particular chimed with Greek culture, which idolized beauty of form and symmetry. Mathematics became a cornerstone of Classical Greek thinking, reflected in its art, architecture, and even philosophy. The almost mystical qualities of geometry and numbers inspired Pythagoras and his followers to establish a cultlike community, dedicated to studying the mathematical principles they believed were the foundations of the Universe and everything in it.

  Centuries before Pythagoras, the Egyptians had used a triangle with sides of 3, 4, and 5 units as a building tool to ensure corners were square. They had come across this idea by observation, and then applied it as a rule of thumb, whereas the Pythagoreans set about rigorously showing the principle, offering a proof that it is true for all right-angled triangles. It is this notion of proof and rigor that is the Greeks’ greatest contribution to mathematics.

  Plato’s Academy in Athens was dedicated to the study of philosophy and mathematics, and Plato himself described the five Platonic solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Other philosophers, notably
Zeno of Elea, applied logic to the foundations of mathematics, exposing the problems of infinity and change. They even explored the strange phenomenon of irrational numbers. Plato’s pupil Aristotle, with his methodical analysis of logical forms, identified the difference between inductive reasoning (such as inferring a rule of thumb from observations) and deductive reasoning (using logical steps to reach a certain conclusion from established premises, or axioms).

  From this basis, Euclid laid out the principles of mathematical proof from axiomatic truths in his Elements, a treatise that was the foundation of mathematics for the next two millennia. With similar rigor, Diophantus pioneered the use of symbols to represent unknown numbers in his equations; this was the first step toward the symbolic notation of algebra.

  A new dawn in the East

  Greek dominance was eventually eclipsed by the rise of the Roman Empire. The Romans regarded mathematics as a practical tool rather than worthy of study. At the same time, the ancient civilizations of India and China independently developed their own numerical systems. Chinese mathematics in particular flourished between the 2nd and 5th centuries CE, thanks largely to the work of Liu Hui in revising and expanding the classic texts of Chinese mathematics.

  IN CONTEXT

  KEY CIVILIZATION

  Babylonians

  FIELD

  Arithmetic

  BEFORE

  40,000 years ago Stone Age people in Europe and Africa count using tally marks on wood or bone.

  6000–5000 BCE Sumerians develop early calculation systems to measure land and to study the night sky.

  4000–3000 BCE Babylonians use a small clay cone for 1 and a large cone for 60, along with a clay ball for 10, as their base-60 system evolves.

  AFTER

  2nd century CE The Chinese use an abacus in their base-10 positional number system.

  7th century In India, Brahmagupta establishes zero as a number in its own right and not just as a placeholder.

  It is given to us to calculate, to weigh, to measure, to observe; this is natural philosophy.

  Voltaire

  French philosopher

  The first people known to have used an advanced numeration system were the Sumerians of Mesopotamia, an ancient civilization living between the Tigris and Euphrates rivers in what is present-day Iraq. Sumerian clay tablets from as early as the 6th millennium BCE include symbols denoting different quantities. The Sumerians, followed by the Babylonians, needed efficient mathematical tools in order to administer their empires.

  What distinguished the Babylonians from neighbors such as Egypt was their use of a positional (place value) number system. In such systems, the value of a number is indicated both by its symbol and its position. Today, for instance, in the decimal system, the position of a digit in a number indicates whether its value is in ones (less than 10), tens, hundreds, or more. Such systems make calculation more efficient because a small set of symbols can represent a huge range of values. By contrast, the ancient Egyptians used separate symbols for ones, tens, hundreds, thousands, and above, and had no place value system. Representing larger numbers could require 50 or more hieroglyphs.

  Using different bases

  The Hindu–Arabic numeration that is employed today is a base-10 (decimal) system. It requires only 10 symbols—nine digits (1, 2, 3, 4, 5, 6, 7, 8, 9) and a zero as a placeholder. As in the Babylonian system, the position of a digit indicates its value, and the smallest value digit is always to the right. In a base-10 system, a two-digit number, such as 22, indicates (2 × 101) + 2; the value of the 2 on the left is ten times that of the 2 on the right. Placing digits after the number 22 will create hundreds, thousands, and larger powers of 10. A symbol after a whole number (the standard notation now is a decimal point) can also separate it from its fractional parts, each representing a tenth of the place value of the preceding figure. The Babylonians worked with a more complex sexagesimal (base-60) number system that was probably inherited from the earlier Sumerians and is still used across the world today for measuring time, degrees in a circle (360° = 6 × 60), and geographic coordinates. Why they used 60 as a number base is still not known for sure. It may have been chosen because it can be divided by many other numbers—1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The Babylonians also based their calendar year on the solar year (365.24 days); the number of days in a year was 360 (6 × 60) with additional days for festivals.

  In the Babylonian sexagesimal system, a single symbol was used alone and repeated up to nine times to represent symbols for 1 to 9. For 10, a different symbol was used, placed to the left of the one symbol, and repeated two to five times in numbers up to 59. At 60 (60 × 1), the original symbol for one was reused but placed further to the left than the symbol for 1. Because it was a base-60 system, two such symbols signified 61, while three such symbols indicated 3,661, that is, 60 × 60 (602) + 60 + 1.

  The base-60 system had obvious drawbacks. It necessarily requires many more symbols than a base-10 system. For centuries, the sexagesimal system also had no place value holders, and nothing to separate whole numbers from fractional parts. By around 300 BCE, however, the Babylonians used two wedges to indicate no value, much as we use a placeholder zero today; this was possibly the earliest use of zero.

  The Babylonian sun-god Shamash awards a rod and a coiled rope, ancient measuring devices, to newly trained surveyors, on a clay tablet dating from around 1000 BCE.

  Other counting systems

  In Mesoamerica, on the other side of the world, the Mayan civilization developed its own advanced numeration system in the 1st millennium BCE—apparently in complete isolation. Theirs was a base-20 (vigesimal) number system, which probably evolved from a simple counting method using fingers and toes. In fact, base-20 number systems were used across the world, in Europe, Africa, and Asia. Language often contains remnants of this system. For example, in French, 80 is expressed as quatre-vingt (4 × 20); Welsh and Irish also express some numbers as multiples of 20, while in English a score is 20. In the Bible, for instance, Psalm 90 talks of a human lifespan being “threescore years and ten” or as great as “fourscore years.”

  From around 500 BCE until the 16th century when Hindu–Arabic numbers were officially adopted in China, the Chinese used rod numerals to represent numbers. This was the first decimal place value system. By alternating quantities of vertical rods with horizontal rods, this system could indicate ones, tens, hundreds, thousands, and more powers of 10, much as the decimal system does today. For example, 45 was written with four horizontal bars representing 4 × 101 (40) and five vertical bars for 5 × 1 (5). However, four vertical rods followed by five vertical rods indicated 405 (4 × 100, or 102) + 5 × 1—the absence of horizontal rods meant there were no tens in the number. Calculations were carried out by manipulating the rods on a counting board. Positive and negative numbers were represented by red and black rods respectively or different cross sections (triangular and rectangular). Rod numerals are still used occasionally in China, just as Roman numerals are sometimes used in Western society.

  The Chinese place value system is reflected in the Chinese abacus (suanpan). Dating back to at least 200 BCE, it is one of the oldest bead-counting devices, although the Romans used something similar. The Chinese version, which is still used today, has a central bar and a varying number of vertical wires to separate ones from tens, hundreds, or more. In each column, there are two beads above the bar worth five each and five beads below the bar worth one each.

  The Japanese adopted the Chinese abacus in the 14th century and developed their own abacus, the soroban, which has one bead worth five above the central bar and four beads each worth one below the bar in each column. Japan still uses the soroban today: there are even contests in which young people demonstrate their ability to perform soroban calculations mentally, a skill known as anzan.

  Cuneiform

  Cuneiform, a word derived from the Latin cuneus (“wedge”) to describe the shape of the symbols, was inscribed into wet clay, stone, o
r metal.

  In the late 1800s, academics deciphered the “cuneiform” (wedge-shaped) markings on clay tablets recovered from Babylonian sites in and around Iraq. Such marks, denoting letters and words as well as an advanced number system, were etched in wet clay with either end of a stylus. Like the Egyptians, the Babylonians needed scribes to administer their complex society, and many of the tablets bearing mathematical records are thought to be from training schools for scribes.

  A great deal has now been discovered about Babylonian mathematics, which extended to multiplication, division, geometry, fractions, square roots, cube roots, equations, and other forms, because—unlike Egyptian papyrus scrolls—the clay tablets have survived well. Several thousand, mostly dating from between 1800 and 1600 BCE, are housed in museums around the world.

  The Babylonian base-60 number system was built from two symbols—the single unit symbol, used alone and combined for numbers 1 to 9, and the 10 symbol, repeated for 20, 30, 40, and 50.

  The Babylonian and Assyrian civilizations have perished…yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in astronomy.