From Here to Infinity: A Guide to Today's Mathematics, a 1996 book by mathematician and science popularizer Ian Stewart

Ian Stewart (mathematician)

Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...

, is a guide to modern mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 for the general reader. It aims to answer questions such as "What is mathematics?", "What is it for " and "What are mathematicians doing nowadays?". Author Simon Singh

Simon Singh

Simon Lehna Singh, MBE is a British author who has specialised in writing about mathematical and scientific topics in an accessible manner....

 describes it as "An interesting and accessible account of current mathematical topics". The first edition, written in 1987, was published under the title The Problems of Mathematics.

Summary

After an introductory chapter The Nature of Mathematics, Stewart devotes each of the following 18 chapters to an exposition of a particular problem that has given rise to new mathematics or an area of research in modern mathematics.

• Chapter 2 - The Price of Primality - primality test
  Primality test
  A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not...
  
  s and integer factorisation
• Chapter 3 - Marginal Interest - Fermat's last theorem
  Fermat's Last Theorem
  In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
  
  
• Chapter 4 - Parallel Thinking - non-Euclidean geometry
  Non-Euclidean geometry
  Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
  
  
• Chapter 5 - The Miraculous Jar - Cantor's theorem
  Cantor's theorem
  In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
  
   and cardinal number
  Cardinal number
  In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
  
  s
• Chapter 6 - Ghosts of Departed Quantities - calculus
  Calculus
  Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
  
   and non-standard analysis
  Non-standard analysis
  Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
  
  
• Chapter 7 - The Duellist and the Monster - the classification of finite simple groups
  Classification of finite simple groups
  In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
  
  
• Chapter 8 - The Purple Wallflower - the four colour theorem
• Chapter 9 - Much Ado About Knotting - topology
  Topology
  Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
  
   and the Poincaré conjecture
  Poincaré conjecture
  In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
  
  
• Chapter 10 - More Ado About Knotting - knot polynomial
  Knot polynomial
  In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.-History:The first knot polynomial, the Alexander polynomial, was introduced by J. W...
  
  s
• Chapter 11 - Squarerooting the Unsquarerootable - complex number
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  s and the Riemann hypothesis
  Riemann hypothesis
  In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
  
  
• Chapter 12 - Squaring the Unsquarable - the Banach-Tarski paradox
• Chapter 13 - Strumpet Fortune - probability
  Probability
  Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
  
   and random walks
• Chapter 14 - The Mathematics of Nature - the stability of the Solar System
  Stability of the Solar System
  The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable historically, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways....
  
  
• Chapter 15 - The Patterns of Chaos - chaos theory
  Chaos theory
  Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
  
   and strange attractors
• Chapter 16 - The Two-and-a-halfth Dimension - fractal
  Fractal
  A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
  
  s
• Chapter 17 - Dixit Algorizmi - algorithm
  Algorithm
  In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
  
  s and NP-complete
  NP-complete
  In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
  
   problems
• Chapter 18 - The Limits of Computability - Turing machine
  Turing machine
  A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
  
  s and computable number
  Computable number
  In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
  
  s
• Chapter 19 - The Ultimate in Technology Transfer - experimental mathematics
  Experimental mathematics
  Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns...
  
  and the relationship between mathematics and science