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‘Algebra and the arithmetic of remainders’ considers a new type of algebra, which is both an ancient topic and one that has found major contemporary application in Internet cryptography. It begins with an outline of abstract algebra, including groups, rings, and fields. Semigroups and groups are algebras with a single associative operation, while rings and fields are algebras with two operations linked via the distributive law. Lattices are algebras with an ordered structure, while vector spaces and modules are algebras where the members can be multiplied by scalar quantities from other fields or rings. The rules of modular arithmetic (or clock arithmetic) and solving linear congruences are also described. ...Less

Matrices represent the central algebraic vehicle for advanced computation throughout mathematics as well as the physical and social sciences. ‘Introduction to matrices’ explains that matrices are simply rectangular arrays of numbers. There are some natural, simple operations that can be performed on matrices. Scalar multiplication is where all entries in a matrix are multiplied by a fixed number. Network theory is one of the major applications of linear algebra, which is the branch of the subject that is largely represented by matrices and matrix calculations. Another application of matrices is to the geometry of transformations. ...Less

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor c of a and b is also a factor of any number of the form ax + by, and since the greatest common divisor (gcd) of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d. ...Less

Applied mathematics is also concerned with the manipulation and analysis of signals and data. In our digital world, where huge amounts of data are routinely collected, transmitted, processed, analysed, compressed, encrypted, decrypted, and stored, efficient mathematical methods and numerical algorithms are needed. ‘Can you picture that? X-rays, DNA, and photos’ outlines how applied mathematics is used to extract information from this data using the examples of computed tomography, X-ray crystallography, and the compression of digital image files. Modern developments have seen the discovery of new methods to extract information and process it efficiently. Examples are the theory of wavelets and compressed sensing, which has been implemented in Magnetic Resonance Imaging scanners. ...Less

Polynomials are expressions of the form p(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ ; the number a_i is the coefficient of x_i , a ₀ is the constant term of p(x), and a_n is the leading coefficient. The underlying algebra of polynomials mirrors that of the integers. Polynomials can be added, subtracted, and multiplied, and the laws of associativity and commutativity and the distributive law of addition over multiplication all hold. Division is more complicated. ‘The algebra of polynomials and cubic equations’ outlines the Remainder and Factor Theorems along with complex numbers in the Argand plane. The factorization of polynomials, the Rational Root Theorem, the Conjugate Root Theorem, and solution of cubic equations are also discussed. ...Less

‘The laws of algebra’ explores the three laws that govern arithmetic operations and explains how these rules are extended so that they continue to be respected as we pass from one number system to a greater one that subsumes the former. The associative law of addition shows that that (a + b) + c = a + (b + c), and the associative law of multiplication is a(bc) = (ab)c. The distributive law tells us how to multiply out the brackets: a(b + c) = ab + ac. The commutative law of addition is a + b = b + a, a law that holds equally well for multiplication: ab = ba. ...Less

How did Leibniz propose to pursue his all-embracing programme of scientific advancement? What were the core projects that held his wide-ranging intellectual life together? ‘Characteristica universalis, logical calculus, and mathematics’ explains that Leibniz nurtured the dream of developing an alphabet of human thoughts leading to the creation of a characteristica universalis: a universal system of signs designed to eliminate the ambiguity of natural language. This project progressed into the development of a logical calculus. Over and above the provision of a means of universal and unambiguous communication, however, the characteristica universalis was conceived by Leibniz as a powerful tool of scientific discovery and judgement on the model of algebra. ...Less