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

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## Patrick J. Murphy and Albert F. Kempf:

## Contents

## 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

## Further Editions

## Source work progress

- 1971: Patrick J. Murphy and Albert F. Kempf:
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