The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and listeners of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought.

Numbers are integral to our everyday lives and factor into almost everything we do. In this Very Short Introduction audiobook, Peter M. Higgins, a renowned popular-science writer, unravels the world of numbers, demonstrating its richness and providing an overview of all the number types that feature in modern science and mathematics.

Quantum theory is the most revolutionary discovery in physics since Newton. This book gives a lucid, exciting, and accessible account of the surprising and counterintuitive ideas that shape our understanding of the sub-atomic world. It does not disguise the problems of interpretation that still remain unsettled 75 years after the initial discoveries. Uncertainty, probabilistic physics, complementarity, the problematic character of measurement, and decoherence are among the many topics discussed.

The study of geometry is at least 2500 years old, and it is within this field that the concept of mathematical proof first arose. To this day geometry remains a very active area of research in mathematics. This Very Short Introduction covers the areas of mathematics falling under geometry, starting with topics such as Euclidean and non-Euclidean geometries, and ranging to curved spaces, projective geometry in Renaissance art, and geometry of space-time inside a black hole. Throughout, Maciej Dunajski outlines the role geometry plays in the broader context of science and art.

Number theory is the branch of mathematics that is primarily concerned with the counting numbers. Of particular importance are the prime numbers, the "building blocks" of our number system. The subject is an old one, dating back over two millennia to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no one has ever been able to resolve them. But number theory has also recently become of great practical importance.

If you move at high speed, time slows down, space squashes up, and you get heavier. Travel fast enough and you could weigh as much as a jumbo jet, be flattened thinner than a CD without feeling a thing - and live forever! As for the angles of a triangle, they do not always have to add up to 180 degrees. And then, of course, there are black holes. These are but a few of the extraordinary consequences of Einstein's theory of relativity.

How much faith should we place in what scientists tell us? Is it possible for scientific knowledge to be fully "objective"? What, really, can be defined as science? In the second edition of this very short introduction, Samir Okasha explores the main themes and theories of contemporary philosophy of science and investigates fascinating, challenging questions such as these.

Games are everywhere: Drivers maneuvering in heavy traffic are playing a driving game. Bargain hunters bidding on eBay are playing an auctioning game. The supermarket's price for corn flakes is decided by playing an economic game. This Very Short Introduction offers a succinct tour of the fascinating world of game theory, a groundbreaking field that analyzes how to play games in a rational way.

In this Very Short Introduction, Simon Horobin investigates how we have arrived at the English we know today and celebrates the way new speakers and new uses mean that it continues to adapt. Engaging with contemporary concerns about correctness, Horobin considers whether such changes are improvements or evidence of slipping standards.

Taking a thematic rather than strictly chronological approach, W. F. Bynum, explores the key turning points in the history of Western medicine - such as the first surgical procedures, the advent of hospitals, the introduction of anesthesia, X-Rays, vaccinations, and many other innovations, as well as the rise of experimental medicine. The book also explores Western medicine's encounters with Chinese and Indian medicine, as well as nontraditional treatments such as homeopathy, chiropractic, and other alternative medicines.

Some people are cleverer than others. This everyday observation is the subject of an academic field that is often portrayed as confused and controversial, when in fact, the field of intelligence holds some of psychology's best-replicated findings. This Very Short Introduction audiobook describes what psychologists have discovered about how and why people differ in their thinking powers. Drawing on largescale data, Ian Deary considers how many types of intelligence there are and how intelligence changes with age.

Bertrand Russell wrote that mathematics can exalt "as surely as poetry". This is especially true of one equation: ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity, or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections.

This very short introduction offers listeners a superb overview of the teachings of the Buddha, as well as a succinct guide to the integration of Buddhism into daily life.

This Very Short Introduction discusses the key concepts of biochemistry, as well as the historical figures in the field and the molecules they studied, before considering the current science and innovations in the field, and the interaction between biochemistry, biotechnology, and synthetic biology.

In this Very Short Introduction, Gerhard L. Weinberg provides an introduction to the origins, course, and impact of the war on those who fought and the ordinary citizens who lived through it. Starting by looking at the inter-war years and the German invasion of Poland in September 1939, he examines how the war progressed by examining a number of key events, including the war in the West in 1940, Barbarossa, the German Invasion of the Soviet Union, the expansion of Japan's war with China, developments on the home front, and the Allied victory from 1944-45.

In this Very Short Introduction audiobook, Jan Zalasiewicz gives a brief introduction to the fascinating field of geology. Describing how the science developed from its early beginnings, he looks at some of the key discoveries that have transformed it before delving into its various subfields, such as sedimentology, tectonics, and stratigraphy.

This is a fascinating introduction to the history of Western astronomy, from prehistoric times to the origins of astrophysics in the mid-19th century. Historical records are first found in Babylon and Egypt, and after two millennia the arithmetical astronomy of the Babylonians merged with the Greek geometrical approach to culminate in the Almagest of Ptolemy. This legacy was transmitted to the Latin West via Islam and led to Copernicus's claim that the Earth is in motion.

This lively and engaging book is the ideal introduction for anyone who has ever been puzzled by what philosophy is or what it is for. Edward Craig argues that philosophy is not an activity born from another planet; learning about it is just a matter of broadening and deepening what most of us do already. He shows that philosophy is no mere intellectual pastime.

Brett L. Walker tackles key themes regarding Japan's relationships with its minorities, state and economic development, and the uses of science and medicine. The book begins by tracing the country's early history through archaeological remains, before proceeding to explore life in the imperial court, the rise of the samurai, civil conflict, encounters with Europe, and the advent of modernity and empire. Walker's vibrant and accessible new narrative then follows Japan's ascension from the ashes of World War II into the thriving nation of today.

Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In The Joy of x, Steven Strogatz expands on his hit New York Times series to explain the big ideas of math gently and clearly, with wit, and insight.

This book is a clear and informative introduction to cryptography and data protection - subjects of considerable social and political importance. It explains what algorithms do, how they are used, the risks associated with using them, and why governments should be concerned. Important areas are highlighted, such as Stream Ciphers, block ciphers, public key algorithms, digital signatures, and applications such as e-commerce.

Here is the extraordinary story of the unfolding of life on Earth, told by Michael J. Benton, a world-renowned authority on biodiversity. Ranging over four billion years, Benton weaves together the latest findings on fossils, earth history, evolutionary biology, and many other fields to highlight the great leaps that enabled life to evolve from microbe to human - big breakthroughs that made whole new ways of life possible - including cell division and multicellularity, hard skeletons, the move to land, the origin of forests, the move to the air.

Ludwig Wittgenstein (1889-1951) was an extraordinarily original thinker, whose influence on 20th-century thinking far outside the bounds of philosophy alone. In this engaging Introduction, A. C. Grayling makes Wittgenstein's thought accessible to the general listener by explaining the nature and impact of Wittgenstein's views. He describes both his early and later philosophy, the differences and connections between them, and gives a fresh assessment of Wittgenstein's continuing influence on contemporary thought.

This Very Short Introduction presents the field of fluid mechanics by focusing on the underlying physical ideas and using everyday phenomena to demonstrate them, from dripping taps to swimming ducks. Eric Lauga shows how this set of fundamental physical concepts can be applied to a wide range of flow behaviors and highlights the role of fluid motion in both the natural and industrial worlds. This book also considers future applications of fluid mechanics in science.

Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greece and bringing us right up to the discovery of gravitational waves. Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes "backwards" sometimes; how to turn the tide in the fight against AIDS.