Mastering Technical Mathematics by Gibilisco, Stan -- Read -- Imperial Library of Trantor

Index

Cover Half Title Title Copyright Dedication 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