Book:Patrick J. Murphy/The New Mathematics Made Simple/Second Edition

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Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd Edition)

Published $\text {1971}$, W.H. Allen, London

ISBN 0 491 00849 X.


Contents

Foreword
$1$ Sets
The empty set
Set equality
Subsets
Numbers
Union of sets
Intersection of sets
Disjoint sets
Universal sets
Complement
Operations. Commutative and associative laws
$2$ Whole Numbers
Operations with whole numbers
Addition
Addition is commutative
Identity number for addition
Addition is associative
A number line
Addition on a number line
Order of whole numbers
Inverse operations
Subtraction
Subtraction on a number line
Properties of subtraction
Multiplication as repeated addition
Identity number of multiplication
Multiplication is associative
The distributive property
Estimating a product
Division
Properties of division
Remainders in division
$3$ The Set of Integers
Closure under addition
The integers
The number line
Order of the integers
Additive inverses
Addition of integers
The meaning of the $`-'$ operation
Multiplication on a number line
Multiplication of integers
Property of negative one
Division of integers
$4$ Solving Equations and Problems
Relation symbols
Grouping symbols
Number sentences
Open sentences
Replacement set
Solution set
Equations
Addition property of equations
Multiplication property of equations
Division property of equations
Solving equations
Translating English phrases
Solving problems
$5$ Rational Numbers
A need for new numbers
Other names for rational numbers
Multiplication of rational numbers
Renaming rational numbers
Mixed numerals in multiplication
Reciprocals or multiplicative inverses
Mixed numerals in addition
Subtraction of rational numbers
Decimals
Decimal numeration
Repeating decimals
Density of rational numbers
$6$ Finite Arithmetics
Clock arithmetic
Additive inverses and subtraction
Multiplicative inverses and division
Modular arithmetic
'Fractions' in finite arithmetic
Other finite systems
Groups
Isomorphism
Sub groups
$7$ Multibase Arithmetic
Conversion to base ten
Conversion from base ten
Rounding off
Addition,
Subtraction
Multiplication
Division
Nim
Recurring bicimals
$8$ Sets of Points
The lines
Line segments and rays
Assumptions about points and lines
Planes
Parallel lines and planes
Separation properties
Simple closed figures
Circles
Angles
Measuring angles
Types of angles
Perpendicular lines and planes
What measurement is
Approximate nature of measurement
Precision
Congruence and similarity
Bisecting line segments and angles
Constructing congruent angles
Perpendicular lines
Congruent triangles
Conditions for congruent triangles
Identity congruence
Vertical angles
Parallel lines and transversals
Proving two triangles congruent
Similar triangles
Angles of a triangle
More about similar triangles
$9$ Relations
Classification of relations
The inverse relation
The graph
Co-ordinate axes
Functions
Inverse relations and functions
Function notation
Composition of functions
$10$ Linear Programming
General equations to the straight line
Inequalities
Maximizing and minimizing
Non-linear programming
$11$ Matrices
Transformations
Reflections
Rotations
Radial expansion
The general transformation
Matrix multiplication
Scalar product
Matrix algebra
Multiplication by a number
Addition
Subtraction
Matrix equations
A group of matrices with respect to multiplication
The inverse $2 \times 2$ matrix
Simultaneous equations
$12$ Transformation Geometry
Reflections
Axis of symmetry
Translations
Vectors
Vector addition
Subtraction
Associativity
Components
Rotations
Rotation of a line
Radial expansion or enlargement
$13$ Elementary Topology
Topological transformations
Networks
Unicursal networks
Networks for maps
Trees
Betti numbers
Surfaces
Classifications of surfaces
The Moebius strip
The punctured torus
The four-colour problem
Answers
Index


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