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Algebra marked the beginning of modern mathematics, moving it beyond arithmetic, which involves calculations featuring given numbers, to problems in which some quantities are unknown. Now, it stands as a pillar of mathematics, underpinning the quantitative sciences. Algebra: A Very Short Introduction explains algebra from scratch. Over the course of ten chapters, it offers a step by step approach for readers keen on developing their understanding of algebra. Using theory and example, it renews the reader’s acquaintance with school mathematics, before taking them progressively further and deeper into modern algebra, including groups, rings, fields, and vector spaces.Less

Combinatorics is the branch of mathematics concerned with selecting, arranging, and listing or counting collections of objects. Dating back some 3000 years, and initially consisting mainly of the study of permutations and combinations, its scope has broadened to include topics such as graph theory, partitions of numbers, block designs, design of codes, and latin squares. Combinatorics: A Very Short Introduction provides an overview of the field and its applications in mathematics and computer theory, considering problems from the shortest routes covering certain stops to the minimum number of colours needed to draw a map with different colours for neighbouring countries.Less

Fractals: A Very Short Introduction looks at the roots of the ‘fractal revolution’ that occurred in mathematics in the 20th century. It presents the ‘new geometry’ of fractals, explains the basic concepts, and explores the wide range of applications in science, and in aspects of economics. Many are familiar with the beauty and ubiquity of fractal forms within nature. Unlike the study of smooth forms, fractal geometry describes more familiar shapes and patterns, such as the complex contours of coastlines, the outlines of clouds, and the branching of trees. This is essential introductory reading for students of mathematics and science, and those interested in popular science and mathematics.Less

The aim of Mathematics: A Very Short Introduction is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. It offers readers an insight into such seemingly paradoxical concepts as infinity, imaginary numbers, and curved space. The first few chapters are concerned with general aspects of mathematical thought and are followed by chapters on more specific topics such as limits and infinity, dimension, geometry, and estimates and approximations. It concludes with some answers to common sociological questions about the mathematical community.Less

Networks: A Very Short Introduction considers the basic elements of network theory and its applications using everyday examples from society, technology, nature, and history. It is impossible to understand the spread of an epidemic, a computer virus, large-scale blackouts, or massive extinctions without taking into account the network structure that underlies all these phenomena. The ubiquitous role of networks; how networks self-organize; why the rich get richer; and how networks can spontaneously collapse are all considered. The findings of complex network theory have very wide and important applications in genetics, ecology, communications, economics, and sociology.Less

Numbers: A very Short Introduction unravels the world of numbers; demonstrating its richness, and providing a comprehensive view of the idea of the number. This VSI paints a picture of the number world, considering how the modern number system matured over centuries. Explaining the various number types and showing how they behave, it introduces key concepts such as integers, fractions, real numbers, and imaginary numbers. By approaching the topic in a non–technical way and emphasising the basic principles and interactions of numbers with mathematics and science, this VSI also demonstrates practical interactions and modern applications, such as encryption of confidential data on the internet.Less

Symmetry: A Very Short Introduction provides an introduction to the formal theory of symmetry: group theory. Now a branch of abstract algebra, this subject first arose in the theory of equations. Symmetry is an immensely important concept in mathematics and throughout the sciences, and its applications range across the entire subject. Symmetry governs the structure of crystals, innumerable types of pattern formation, how systems change their state as parameters vary; and fundamental physics is governed by symmetries in the laws of nature. It is highly visual, with applications that include animal markings, locomotion, evolutionary biology, elastic buckling, waves, the shape of the Earth, and the form of galaxies.Less

Topology, the mathematical study of the properties that are preserved through the deformations, twistings, and stretchings of objects, is an important area of modern mathematics. Topology: A Very Short Introduction provides a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eye-opening examples that led mathematicians to recognize a need for studying topology, it pays homage to the historical people, problems, and surprises that have propelled the growth of this field. As broad and fundamental as algebra and geometry, its study has important implications for science more generally, especially physics.Less

Trigonometry: A Very Short Introduction draws together the full history of trigonometry, stretching across two millennia and several cultures such as ancient Greece, medieval India, and the Islamic world. It introduces the key concepts of trigonometry, drawing readers beyond the basic relationships first encountered in school to reveal the richness of the entire subject of trigonometry and ideas such as curved space. It also explores connections with genuine modern applications, including navigation, the analysis of music, computer graphics, and powerful modelling tools in science, and shows how trigonometry has participated in big questions about the world, including the shape of the universe and the nature of infinity.Less