Preface

Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ ended with the real numbers as a complete ordered field. This volume supplies what both analysis and algebra use on their first pages without stopping to prove it. It has three parts. Part I, common ground (Chapters 1–3), holds what both branches stand on: finite sums, the binomial theorem, inequalities and real powers; the arithmetic of the integers, divisibility and congruences; and the complex numbers, built on those real numbers. Part II, algebra (Chapters 4–7), builds what linear algebra and the algebra volumes assume: fields as objects of their own, groups through permutations, polynomials as formal objects, and the arithmetic of polynomials over a field, ending with the Fundamental Theorem of Algebra. Part III, tools of analysis (Chapters 8–10), builds what analysis assumes: real functions and their graphs, coordinates and vectors in $\mathbb{R}^n$ , and lines, hyperplanes and convex sets.

Who it is for. The reader we have in mind has worked through Logic and Proof, Sets and Functions and the main line of Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , or has their habits from elsewhere. That means running an induction, checking that an operation on classes is well defined, and proving a statement about every element of a ring from the ring axioms alone. The chapters point back to those volumes wherever they use a definition or a result, with a link, so a reader who is unsure of a step can look it up. Much of the material is familiar in outline from school or from a first linear algebra course; what is new is that every rule is proved, and that nothing is used before it is proved.

What it covers.

Chapter	Title	What it does
1	Inequalities, Finite Sums, and Real Powers	Finite sums and telescoping, the binomial theorem in a commutative ring, the strict Bernoulli inequality and the arithmetic–geometric mean inequality, estimates for limits, real $n$ -th roots and rational powers
2	Divisibility and Congruences	Bézout’s theorem and the Euclidean algorithm, Euclid’s lemma and unique prime factorization, ideals and quotient rings, $\mathbb{Z}/n\mathbb{Z}$ as a commutative ring, units, $\mathbb{Z}/p\mathbb{Z}$ a field, and the Chinese remainder theorem
3	Complex Numbers	$\mathbb{C}$ as pairs of real numbers, proved a field containing $\mathbb{R}$ ; conjugate, modulus and the triangle inequality; square roots and the quadratic formula; why $\mathbb{C}$ has no order; polar form, De Moivre’s formula and the roots of unity
4	Fields	The field axioms and their consequences; the finite fields $\mathbb{F}_p$ ; subfields and the subfield test; field homomorphisms; the characteristic and the prime subfield; the Frobenius map and Fermat’s little theorem
5	Groups, Subgroups, and Homomorphisms	Groups, products of many factors and integer powers; subgroups and the subgroups of $\mathbb{Z}$ ; homomorphisms, kernels and isomorphisms; permutations, cycle notation and the disjoint cycle decomposition; transpositions generate the symmetric group
6	Formal Polynomials	A polynomial as a sequence of coefficients; the ring $R[x]$ over a commutative ring; degree and integral domains; evaluation and substitution as ring homomorphisms, and why a polynomial is not its function; the formal derivative
7	Polynomials over a Field	Division with remainder, the greatest common divisor and the Euclidean algorithm in $F[x]$ ; irreducible polynomials and unique factorization; roots, multiplicity and repeated roots; interpolation, the rings $F[x]/\langle m \rangle$ and the Chinese remainder theorem; the Fundamental Theorem of Algebra and the factorization of real polynomials
8	Real Functions and Their Graphs	Real functions, their graphs and their combinations; monotone functions and inverses; bounds and suprema of functions; floor and step functions; the exponential, the logarithm and real powers; a table of strange functions
9	Coordinates and Vectors in $\mathbb{R}^n$	Points and vectors in $\mathbb{R}^n$ ; the dot product, length and distance; the Cauchy–Schwarz and triangle inequalities; projection onto a line and the cosine of an angle
10	Lines, Hyperplanes, and Convex Sets	Lines and segments, and betweenness by distances; hyperplanes and nearest points; convex sets and convex hulls

Ways through. Read Chapters 1–3 first; each uses the ones before it. A reader heading for linear algebra and algebra then reads Chapters 4–7 in order. Of Chapter 1 that reader needs Sections 1.1 and 1.2 (finite sums, natural multiples and the binomial theorem), which Chapter 4 recalls; Sections 1.3–1.5 can wait. A reader heading for analysis reads all of Chapter 1, then Chapter 3 up to Section 3.5, then Chapters 8–10 in order. Chapters 8–10 use the algebra of Part II only in a few places: polynomial functions and the root count of Chapters 6 and 7 (Sections 8.1, 8.3 and 8.6), lowest terms from Chapter 2 and a subgroup test from Chapter 5 (Section 8.6), and the complex numbers of Chapter 3 (Sections 9.4–9.6). Every section has Quick checks with folded answers; answer each one before you unfold it. The exercises of a section come in three groups, A to check understanding, B for practice and C to go deeper, and the last section of each chapter adds R items that reach back to earlier chapters. A chapter’s opening page lists what it needs, with links, and the time it takes. Second-reading folds hold edge cases and longer justifications. They can stay closed on a first reading, but they are not optional, since later proofs may use what they contain.

Two borrowed facts. This volume does not define cosine, sine, $\pi$ or the exponential function: defining them needs limits and power series, which belong to analysis. Section 3.5 states five facts about cosine, sine and $\pi$ , numbered (T1)–(T5), as an assumption (Axiom 3.5.2), and Section 8.5 states three facts about the exponential function, numbered (E1)–(E3) (Axiom 8.5.1). The analysis volumes prove both groups. The first is used only for polar form and for the roots of complex numbers in Sections 3.5 and 3.6, and the second only in Section 8.5 below its box. Each proof that uses one of the facts names it. Chapters 4–7 use (T1)–(T5) only in examples and remarks, and Chapter 7 proves the Fundamental Theorem of Algebra from the square roots of Section 3.4 and the completeness of $\mathbb{R}$ alone. Chapters 1, 9 and 10 use neither group, apart from a comparison in the Looking-ahead box of Section 9.6.

Time. The chapters add up to about 240 hours if you read the main text and do every A exercise and one B and one C exercise per section, and about 350 hours for everything. By chapter, the two numbers are: Chapter 1, 20 and 30; Chapter 2, 20 and 30; Chapter 3, 25 and 35; Chapter 4, 25 and 40; Chapter 5, 25 and 35; Chapter 6, 25 and 35; Chapter 7, 30 and 45; Chapter 8, 30 and 45; Chapter 9, 25 and 35; Chapter 10, 15 and 20. On the first measure, the algebra route (Sections 1.1 and 1.2 and Chapters 2–7) takes about 160 hours, and the analysis route (Chapters 1, 3 and 8–10) about 115.

What comes next. Analysis on the Real Line and Integration, Series, and Function Spaces use Chapters 1, 3 and 8–10. The linear algebra volumes, beginning with Spaces and Maps, take their scalars from a field and use the complex numbers, polynomials and their roots throughout: Chapter 4’s fields and subfields, Chapter 6’s evaluation of a polynomial, Chapter 7’s division with remainder and the Fundamental Theorem of Algebra, and Chapter 5’s permutations, which Determinants and the Structure of Operators needs for the determinant. The algebra volumes start where Chapter 7 stops. Groups continues Chapter 5 with cyclic groups, the sign of a permutation, cosets and Lagrange’s theorem; Rings and Fields continues Chapters 4, 6 and 7 with fraction fields, polynomials over the integers, Gauss’s lemma and field extensions. Many of their first definitions are the ones proved here, under the same names.