Mastering Technical Mathematics, Third Edition

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Mastering Technical Mathematics, Third Edition

Mastering Technical Mathematics, Third Edition

Authors: Stan Gibilisco and Norman Crowhurst

Published: October 2007

eISBN: 9780071595421 0071595422 | ISBN: 9780071494489

Contents

Introduction

Part 1 Working with Numbers

1 From Counting to Addition

Numbers and Numerals

Counting in Tens and Dozens

Writing Numerals Greater Than 10

Why Zero Is Used in Counting

By Tens and Hundreds to Thousands

Don’t Forget the Zeros

Millions and More

What About Infinity?

Addition Is Counting on

Adding Three or More Numbers

Adding Larger Numbers

Carrying

Checking Answers

Weights

Using the Grocer’s Balance

Questions and Problems

2 Subtraction

Taking Away

Checking Subtraction by Addition

Borrowing

Subtracting with Larger Numbers

Working with Money

Making Change

Subtracting Weights

Subtracting Liquid Measures

Questions and Problems

3 Multiplication

It’s a Shortcut

Making a Table

Patterns in Multiplication Tables

How Calculators Multiply

Carrying in Multiplication

Using Your Calculator to Verify This Process

Put Those Zeros to Work!

Using Subtraction to Multiply

Multiplying by Factors

Multiplying in Nondecimal Systems

Questions and Problems

4 Division

Let’s Think Like a Calculator

Division “Undoes” Multiplication

Alternative Notation

Dividing into Longer Numbers

Multiplication Checks Division

More About How a Calculator Does It

Dividing by Larger Numbers (The People Way)

Division by Factors

When a Remainder Is Left

How a Calculator Handles Fractions

Decimal Equivalents of Fractions

More Difficult Fractions

Where Groups of Digits Repeat

Converting Recurring Decimals to Fractions

Questions and Problems

5 Fractions

Slicing up a Pie

Factors Help Simplify Fractions

Spotting the Factors

Prime Numbers

Factoring into Primes

Prime Factors of 139

Prime Factors of 493

Prime Factors of 546

Adding and Subtracting Fractions

Finding the Common Denominator

Significant Figures

Approximate Division and Multiplication

Approximate Addition and Subtraction

Questions and Problems

6 Plane Polygons

Length Times Length Is Area

What Is a Square?

Different Shapes with the Same Area

Square Measure Vs. Linear Measure

Right Triangles

Parallelograms

Area of a Parallelogram

Area of an Acute Triangle

Area of an Obtuse Triangle

Metric Measure

Area Problems

Questions and Problems

7 Time, Percentages, and Graphs

Time as a Dimension

Time and Speed

Average Speed

Making up for Lost Time

Fractional Growth

Percentages

Percentages with Money

Percentages up and down

The Bar Graph

Other Graphs

Interpolation

Graphs Can Help You Evaluate Data and Find Errors

Questions and Problems

Part 2 Algebra, Geometry, and Trigonometry

8 First Notions in Algebra

Shortcuts for Long Problems

Graphs Help You Find Errors and Solutions

Manipulating Equations

Writing It as Algebra

Italics—or Not?

Grouping Symbols

Solving Problems with Grouping Symbols

Putting a Problem into Algebra

Magic by Algebra

A Boat-in-the-River Problem

Number Problems

Questions and Problems

9 Some “School” Algebra

Orderly Writing in Algebra

Showing “Place” in Algebra

Dimension in Algebra

Equations and Inequalities

A Consecutive-Number Problem

Simultaneous Equations

A Fraction Problem

A Little More Practice

Solving by Substitution

Solving for Reciprocals

Long Division and Factoring

Questions and Problems

10 Quadratic Equations

A Quadratic Example

Linear Vs. Quadratic Forms

A Quadratic Graph Is Symmetric

Solving a Quadratic Equation

Factors of a Quadratic

Finding Factors

Completing the Square

What the Answers Mean

The Quadratic Formula

A Quadratic Problem

Questions and Problems

11 Some Useful Shortcuts

Sum and Difference in Geometry

Difference of Squares Finds Factors

Finding Square Roots

Signs in Successive Roots

Real Vs. Imaginary Numbers

Complex Numbers

Complex Numbers Find New Roots

A Problem Using Simultaneous Equations

Questions and Problems

12 Mechanical Mathematics

What Is Force?

Units of Force

Acceleration, Speed, and Distance

Force and Work

Work and Energy

Energy and Power

Gravity as a Source of Energy

Weight as Force

Gravitational Measure of Work

Energy for Constant Acceleration

Potential and Kinetic Energy

Kinetic Energy and Speed

Acceleration at Constant Power

How a Spring Stores Energy

How a Spring Transfers Energy

Resonance Cycle

Travel and Speed in Resonant System

Questions and Problems

13 Ratio and Proportion

Fractions, Projections, Extremes, and Means

Proportions in Practice

Shape and Size

About Angles in Triangles

Use of Right Triangles

Angles Identified by Ratios

Theorem of Pythagoras

Names for Angle Ratios

Spotting the Triangle

Degree Measure of Angles

Three Applications of Trigonometry

Questions and Problems

14 Trigonometric and Geometric Calculations

Ratios for Sum Angles

Finding Tan (A + B)

An Example: Ratios for 75°

Ratios for Angles Greater Than 90°

Ratios for Difference Angles

Ratios Through the Four Quadrants

Pythagoras in Trigonometry

Double Angles

Higher Multiple Angles

Properties of the Isosceles Triangle

Angles in a Circle

Angles in a Semicircle

Definitions

Questions and Problems

Part 3 Analysis and Calculus

15 Systems of Counting

Mechanical Counters

Duodecimal Numbers

Conversion from Decimal to Duodecimal

Conversion from Duodecimal to Decimal

Octal Numbers

Hexadecimal numbers

Binary Numbers

Comparing the Values

Converting Decimal to Binary

Binary Multiplication

Binary Division

Indices

Negative and Fractional Exponents

Surds

Questions and Problems

16 Progressions, Permutations, and Combinations

Arithmetic Progression

Geometric Progression

Harmonic Progression

Arithmetic Series

Geometric Series

Convergence

Rate of Convergence

Permutations

Combinations

Powers of a Binomial

Alternative Notation for Multiplying Numbers

Questions and Problems

17 Introduction to Derivatives

Rates of Change

Infinitesimal Changes

Successive Differentiation

Differentiating a Polynomial

Successive Differentiation of Movement

Circular Measure of Angles

Derivatives of Sine and Cosine

Successive Derivatives of Sine

Questions and Problems

18 More About Differentiation

Frequency and Period of a Sine Wave

Sinusoidal Motion

Harmonic Motion

Linear and Nonlinear Relationships

Multiples and Powers

Functions

Derivative of Constant

Derivative of Sum of Two Functions

Derivative of Difference of Two Functions

Derivative of Function Multiplied by a Constant

Derivative of Product of Two Functions

Derivative of Product of Three Functions

Derivative of Quotient of Two Functions

Reciprocal Derivatives

Derivative of Function Raised to a Power

Chain Rule

Questions and Problems

19 Introduction to Integrals

Adding up the Pieces

The Antiderivative

Patterns in Calculations

The Constant of Integration

Definite Integrals

Finding Area by Integration

Area of a Circle

Curved Areas of Cylinders and Cones

Surface Area of a Sphere

Volumes of Wedges and Pyramids

Volumes of Cones and Spheres

Questions and Problems

20 Combining Calculus with Other Tools

Local Maximum and Minimum

Point of Inflection

Maximum and Minimum Slope

Maximum Area with Constant Perimeter

Box with Minimum Surface Area

Cylinder with Minimum Surface Area

Cone with Minimum Surface Area

Equations for Circles, Ellipses, and Parabolas

Directrix, Focus, and Eccentricity

The Ellipse and the Circle

Relationships Between Focus, Directrix, and Eccentricity

Focus Property of Parabola

Focus Property of Ellipse

Hyperbola: Eccentricity Greater Than 1

Geometry of Conic Sections

Questions and Problems

21 Alternative Coordinate Systems

Mathematician’s Polar Coordinates

Some Examples

Compressed Polar Coordinates

Mathematician’s Polar vs. Cartesian Coordinates

Navigator’s Polar Coordinates

Mathematician’s Vs. Navigator’s Polar Coordinates

Navigator's Polar Vs. Cartesian Coordinates

Latitude and Longitude

Celestial Coordinates

Cartesian Three-Space

Cylindrical Coordinates

Spherical Coordinates

Questions and Problems

22 Complex Numbers

Imaginary Quantities

The Complex Number Plane

Multiplying Complex Quantities

Three Cube Root Examples

Reciprocal of a Complex Quantity

Division of Complex Quantities

Rationalization

Checking Results

Operation Summary

Use of a Complex Plane

Roots by Complex Quantities

Questions and Problems

Part 4 Tools of Applied Mathematics

23 Trigonometry in the Real World

Parallax

The Base-line

Accuracy

Distance to an Object

Stadimetry

Interstellar Distance Measurement

Distance Vs. Parallax

A Point of Confusion

Radar

Laws of Sines and Cosines

Use of Radar

How to Hunt a "Fox"

The Fox Finds Itself

Position of a Ship

Questions and Problems

24 Logarithms and Exponentials

The Logarithm Base

Common Logarithms

Natural Logarithms

Basic Properties of Logarithms

Conversion of Log Functions

Logarithm Comparisons

The Exponential Base

Common Exponentials

Natural Exponentials

Alternative Expressions

Logarithms Vs. Exponentials

Basic Properties of Exponentials

Exponential Comparisons

Numerical Orders of Magnitude

Questions and Problems

25 Scientific Notation Tutorial

Standard Form

Alternative Form

The "Times" Sign

Plain-Text Exponents

Orders of Magnitude

When to Use Scientific Notation

Prefix Multipliers

Multiplication

Division

Raising to a Power

Taking a Root

Addition

Subtraction

Significant Figures

How Accurate Are You?

What About 0?

What About Exact Values?

More About Addition and Subtraction

Questions and Problems

26 Vectors

Magnitude and Direction

Sum of Two Vectors

Multiplication of a Vector by a Scalar

Dot Product of Two Vectors

Magnitude and Direction in the Polar Plane

Sum of Two Vectors in the Polar Plane

Multiplication of a Vector by a Scalar in the Polar Plane

Dot Product of Two Vectors in Polar Plane

Magnitude in XYZ Space

Direction in XYZ Space

Sum of two Vectors in XYZ Space

Multiplication of a Vector by a Scalar in XYZ Space

Dot Product of Two Vectors in XYZ Space

Cross Product of Two Vectors

Negative Vector Magnitudes

Questions and Problems

27 Logic and Truth Tables

Sentences

Negation (Not)

Conjunction (And)

Disjunction (Or)

Implication (If/Then)

Logical Equivalence (Iff)

Truth Tables

Tables for Basic Logic Functions

The Equals Sign

Q.E.D

Precedence

Contradiction and Double Negation

Commutative Laws

Associative Laws

Law of Implication Reversal

De Morgan's Laws

Distributive Law

Truth Table Proofs

Proofs Without Tables

Questions and Problems

28 Beyond Three Dimensions

Imagine That!

Time-Space

Position and Motion

Time as Displacement

Universal Speed

The Four-Cube

The Rectangular Four-Prism

Impossible Paths

General Time-Space Hypervolume

Cartesian N-Space

Distance Formulas

Vertices of a Tesseract

Hypervolume of a Four-Prism

Euclid's Axioms

The Parallel Postulate

Geodesics

Modified Parallel Postulate

No Parallel Geodesics

More Than One Parallel Geodesic

Curved Space

The "Hyperfunnel"

Questions and Problems

29 A Statistics Primer

Experiment

Variable

Population

Sample, Event, and Census

Random Variable

Frequency

Parameter

Statistic

Distribution

Frequency Distribution

Cumulative Frequency Data

Mean

Median

Mode

Questions and Problems

30 Probability Basics

Sets and Set Notation

Event Vs. Outcome

Sample Space

Mathematical Probability

Empirical Probability

Law of Large Numbers

Independent Outcomes

Mutually Exclusive Outcomes

Complementary Outcomes

Nondisjoint Outcomes

Multiple Outcomes

The Density Function

Expressing the Pattern

Area under the Curve

Uniform Distribution

Normal Distribution

The Empirical Rule

Questions and Problems

Final Exam

Appendices

1 Solutions to End-of-Chapter Questions and Problems

2 Answers to Final Exam Questions

3 Table of Derivatives

4 Table of Indefinite Integrals

Suggested Additional Reading

Index