Borovik, Alexandre.
The Essence of Mathematics Through Elementary Problems. - 1 online resource (389 pages) - OBP Series in Mathematics Ser. .
Intro -- Preface -- About this text -- I. Mental Skills -- Mental arithmetic and algebra -- Times tables. -- Squares, cubes, and powers of 2. -- Primes -- Common factors and common multiples -- The Euclidean algorithm -- Fractions and ratio -- Surds -- Direct and inverse procedures -- Factorisation -- Structural arithmetic -- Pythagoras' Theorem -- Pythagoras' Theorem, trig for special angles, and CAST -- Converses and Pythagoras' Theorem -- Pythagorean triples -- Sums of two squares -- Visualisation -- Trigonometry and radians -- Sine Rule -- Radians and spherical triangles -- Polar form and sin(A+B) -- Regular polygons and regular polyhedra -- Regular polygons are cyclic -- Regular polyhedra -- Chapter 1: Comments and solutions -- II. Arithmetic -- Place value and decimals: basic structure -- Order and factors -- Standard written algorithms -- Divisibility tests -- Sequences -- Triangular numbers -- Fibonacci numbers -- Commutative, associative and distributive laws -- Infinite decimal expansions -- The binary numeral system -- The Prime Number Theorem -- Chapter 2: Comments and solutions -- III. Word Problems -- Twenty problems which embody ``3 - 1 = 2'' -- Some classical examples -- Speed and acceleration -- Hidden connections -- Chapter 3: Comments and solutions -- IV. Algebra -- Simultaneous linear equations and symmetry -- Inequalities and modulus -- Geometrical interpretation of modulus, of inequalities, and of modulus inequalities -- Inequalities -- Factors, roots, polynomials and surds -- Standard factorisations -- Quadratic equations -- Complex numbers -- Cubic equations -- An extra -- Chapter 4: Comments and solutions -- V. Geometry -- Comparing geometry and arithmetic -- Euclidean geometry: a brief summary -- Areas, lengths and angles -- Regular and semi-regular tilings in the plane. Ruler and compasses constructions for regular polygons -- Regular and semi-regular polyhedra -- The Sine Rule and the Cosine Rule -- Circular arcs and circular sectors -- Convexity -- Pythagoras' Theorem in three dimensions -- Loci and conic sections -- Cubes in higher dimensions -- Chapter 5: Comments and solutions -- VI. Infinity: recursion, induction, infinite descent -- Proof by mathematical induction I -- `Mathematical induction' and `scientific induction' -- Proof by mathematical induction II -- Infinite geometric series -- Some classical inequalities -- The harmonic series -- Induction in geometry, combinatorics and number theory -- Two problems -- Infinite descent -- Chapter 6: Comments and solutions.
9781783747016
Mathematics-Miscellanea.
Electronic books.
QA99 .B676 2019
510
The Essence of Mathematics Through Elementary Problems. - 1 online resource (389 pages) - OBP Series in Mathematics Ser. .
Intro -- Preface -- About this text -- I. Mental Skills -- Mental arithmetic and algebra -- Times tables. -- Squares, cubes, and powers of 2. -- Primes -- Common factors and common multiples -- The Euclidean algorithm -- Fractions and ratio -- Surds -- Direct and inverse procedures -- Factorisation -- Structural arithmetic -- Pythagoras' Theorem -- Pythagoras' Theorem, trig for special angles, and CAST -- Converses and Pythagoras' Theorem -- Pythagorean triples -- Sums of two squares -- Visualisation -- Trigonometry and radians -- Sine Rule -- Radians and spherical triangles -- Polar form and sin(A+B) -- Regular polygons and regular polyhedra -- Regular polygons are cyclic -- Regular polyhedra -- Chapter 1: Comments and solutions -- II. Arithmetic -- Place value and decimals: basic structure -- Order and factors -- Standard written algorithms -- Divisibility tests -- Sequences -- Triangular numbers -- Fibonacci numbers -- Commutative, associative and distributive laws -- Infinite decimal expansions -- The binary numeral system -- The Prime Number Theorem -- Chapter 2: Comments and solutions -- III. Word Problems -- Twenty problems which embody ``3 - 1 = 2'' -- Some classical examples -- Speed and acceleration -- Hidden connections -- Chapter 3: Comments and solutions -- IV. Algebra -- Simultaneous linear equations and symmetry -- Inequalities and modulus -- Geometrical interpretation of modulus, of inequalities, and of modulus inequalities -- Inequalities -- Factors, roots, polynomials and surds -- Standard factorisations -- Quadratic equations -- Complex numbers -- Cubic equations -- An extra -- Chapter 4: Comments and solutions -- V. Geometry -- Comparing geometry and arithmetic -- Euclidean geometry: a brief summary -- Areas, lengths and angles -- Regular and semi-regular tilings in the plane. Ruler and compasses constructions for regular polygons -- Regular and semi-regular polyhedra -- The Sine Rule and the Cosine Rule -- Circular arcs and circular sectors -- Convexity -- Pythagoras' Theorem in three dimensions -- Loci and conic sections -- Cubes in higher dimensions -- Chapter 5: Comments and solutions -- VI. Infinity: recursion, induction, infinite descent -- Proof by mathematical induction I -- `Mathematical induction' and `scientific induction' -- Proof by mathematical induction II -- Infinite geometric series -- Some classical inequalities -- The harmonic series -- Induction in geometry, combinatorics and number theory -- Two problems -- Infinite descent -- Chapter 6: Comments and solutions.
9781783747016
Mathematics-Miscellanea.
Electronic books.
QA99 .B676 2019
510
