Basic Mathematics

Welcome

Welcome to the MAST Basic Mathematics Homepage. On this page you will find links to the lessons for this course, a listing of available papers submitted by students of this course, and a list of references that were useful in making this course.

Introduction

This course has been changed. The new course is now composed of four shorter courses

R001: Basic Mathematics

Description: The topic list for this project is: basic arithmetic operations, basic algebraic manipulation, introduction to number theory, polynomial and rational expressions, and equations and inequalities.

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Project Objectives:

• Review and perfect your ability to perform arithmetic operations.
• To develop a significant level of skill at algebraic manipulations.
• To gain familiarity with fundamental ideas of number theory.
• To gain skill in deriving and solving linear and quadratic equations and inequalities.

Modules:

1. Introduction and basic arithmetic operations.
2. Basic algebraic maniulations.
3. Introduction to number theory.
4. Polynomial and rational expressions.
5. Equations and inequalities.

Tasks for Module #1: Introduction and basic arithmetic operations.

At one topic per day, this module can be completed in 6 weeks.

1. Begin a notebook for the project. This will be worth 1 point upon completion.
2. Natural numbers and counting.
3. Addition.
4. Subtraction.
5. Integers, and addition and subtraction of integers.
6. Multiplication of natural numbers.
7. Multiplication of integers.
8. Use of parentheses to control the order of operations.
9. Division of natural numbers.
10. Division of integers.
11. Rational numbers (fractions).
12. Division and rational numbers.
13. Exponentiation of natural numbers.
14. Exponentiation of integers.
15. Exponentiation by integers.
16. Exponentiation of rational numbers.
17. Decimals and addition and subtraction of decimals.
18. Multiplication of decimals.
19. Division of decimals.
20. Exponentiation of decimals.
21. Square roots.
22. Cube roots.
23. Higher roots and exponentiation by rational numbers.
24. Irrational numbers and real numbers.
25. Imaginary numbers and complex numbers.
26. Logarithms.
27. Ratio.
28. Proportion.
29. Denominate numbers.
30. Percentage.

Tasks for Module #2: Basic algebraic maniulations.

At one topic per day, this module can be completed in 2 weeks.

1. Using non-numerals to denote numbers.
2. Properties of natural numbers.
3. Properties of integers.
4. Properties of rational numbers.
5. Properties of real numbers.
6. Properties of imaginary numbers and complex numbers.
7. Manipulations involving addition and subtraction.
8. Multiplication and division.
9. Exponentiation and root extraction.
10. Logarithm extraction.

Tasks for Module #3: Introduction to number theory.

At one topic per day, this module can be completed in 2 weeks.

1. Divisibility.
2. The division algorithm.
3. Prime numbers.
4. Factorization into primes.
5. Greatest common denominator.
6. The Euclidean algorithm.
7. Least common multiple.
8. Prime divisor property.
9. Unique prime factorization.
10. Perfect numbers, amicable numbers, and Mersenne primes.

Tasks for Module #4: Polynomial and rational expressions.

At one topic per day, this module can be completed in 3 weeks.

1. Monomial and polynomial expressions.
2. Polynomial addition and subtraction.
3. Polynomial multiplication.
4. Polynomial division.
5. Polynomial exponentiation.
6. The binomial theorem.
7. Binomial coefficients.
8. Polynomial factoring.
9. Root extraction of a polynomial.
10. Rational expressions.
11. Rational addition and subtraction.
12. Rational multiplication and division.
13. Rational exponentiation.
14. Integer exponents.
15. Rational exponents.

Tasks for Module #5: Equations and inequalities.

At one topic per day, this module can be completed in 4 weeks.

1. Equations and solutions.
2. Inequalities and solutions.
3. Linear equations.
4. Linear inequalities.
5. Cartesian coordinates.
6. Graphing linear equations and inequalities.
7. Properties of the graph of a linear equation.
8. Quadratic equations.
9. Graphing quadratic equations.
10. Solution by factoring.
11. Roots of a quadratic equation.
12. Completing the square.
13. The quadratic formula.
14. Complex roots.
15. Quadratic inequalities.
16. Complete cubic equation.
17. Graphing the cubic equation.
18. Roots of the reduced cubic equation.
19. Solution of the complete cubic equation.
20. The complete quartic equation.

R002: Geometry

The topic list for this project is: basic geometric ideas and proof methods, the geometry of plane figures and area, congruence and similarity of plane figures, plane constructions, and an introduction to spatial geometry.

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Project Objectives:

• To master the basic ideas of geometry.
• To develop facility in the reading and development of proofs.
• To understand plane figures and their properties.
• To understand the notion of area.
• To understand and apply the principles of congruence and similar figures.
• To be able to construct plane figures using only a straightedge and compass.
• To gain familiarity with spatial figures such as cubes, prisms, spheres, and cones.

Modules:

1. Introduction, basic ideas of geometry, and methods of proof.
2. Plane figures and area.
3. Congruence and similar figures.
4. Construction of plane figures.
5. Introduction to spatial geometry.

Tasks for Module #1: Introduction, basic ideas of geometry, and methods of proof.

At one topic per day, this module can be completed in 4 weeks.

1. Begin a notebook for the project. This will be worth 1 point upon completion.
2. Primitive concepts of point, line, and plane.
3. Propositions.
4. Logical connectives.
5. Truth tables.
6. Tautologies.
7. Rules of logic.
8. Direct proofs.
9. Contrapositive proofs.
10. Reductio ad absurdum proofs.
11. Definitions.
12. Some definitions from points and lines.
13. Line segments.
14. Rays.
15. Betweenness.
16. Midpoint.
17. Intersecting lines.
18. Angles.
19. Right angles.
20. Degrees and angle measurement.

Tasks for Module #2: Plane figures and area.

At one topic per day, this module can be completed in 4 weeks.

1. Triangles.
2. Equilateral triangles and isosceles traingles.
3. Acute triangles and obtuse triangles.
4. Right triangles.
5. Quadrilaterals.
6. Parallelograms and trapezoids.
7. Rectangles.
8. Rhombus.
9. Pentagon.
10. Hexagon.
11. Other polygons.
12. The circle.
13. Segments of a circle.
14. Intersection of a lines with circles.
15. The notion of area.
16. Area of a rectangle.
17. Area of a triangle.
18. Area of a polygon.
19. Area of a circle and a segment of a circle.
20. Pythagorean theorem.

Tasks for Module #3: Congruence and similar figures.

At one topic per day, this module can be completed in 2 weeks.

1. Congruence and congruent triangles.
2. Parallelism.
3. Perpendicularity and triangle congruence.
4. Congruent quadrilaterals.
5. Congruent polygons.
6. Circles and congruence.
7. Similarity and similar triangles.
8. Similar quadrilaterals.
9. Similar polygons.
10. Circles and similarity.

Tasks for Module #4: Construction of plane figures.

At one topic per day, this module can be completed in 2 weeks.

1. Perpendiculars.
2. Parallels.
3. Angles.
4. Triangles.
5. Quadrilaterals.
6. Polygons.
7. Circles.
8. Similar figures.
9. Inscribed figures.
10. Circumscribed figures.

Tasks for Module #5: Introduction to spatial geometry.

At one topic per day, this module can be completed in 2 weeks.

1. Points, lines and planes in space.
2. Intersections of lines and planes in space.
3. Tetrahedrons, pyramids, and prisms.
4. Quadrahedrons, parallelepipeds, and cuboids.
5. Polyhedra and regular polyhedra.
6. Cones.
7. Cylinders.
8. Spheres.
9. Volume.
10. Surface area.

R003 Algebra

The topic list for this project is: set theory, systems of equations, relations and functions, algebraic functions, and exponential and logarithmic functions.

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Project Objectives:

• To master the basic ideas of set theory.
• To derive and solve systems of linear equations.
• To gain familiarity with the notion and basic operations of matrices and determinants.
• To understand relations and equivalence relations.
• To understand functions from a set theoretic point of view.
• To be able to plot the graph of a function.
• To recognize linear, quadratic, cubic, quartic, polynomial, rational, exponential, and logarithmic functions and their inverses.
• To become familiar with the principle theorems of polynomial functions.
• To become familiar with the principle theorems of rationall functions.
• To become familiar with the principle theorems of exponential functions.
• To become familiar with the principle theorems of logarithmic functions.

Modules:

1. Introduction and set theory.
2. Linear systems.
3. Relations and functions.
4. Algebraic functions.
5. Transcendental functions.

Tasks for Module #1: Introduction and set theory.

At one topic per day, this module can be completed in 3 weeks.

1. Begin a notebook for the project. This will be worth 1 point upon completion.
2. Sets and elements; and equal sets.
3. Subsets and proper subsets; the universe of discourse and the empty set.
4. Union.
5. Intersection.
6. Difference.
7. Complement.
8. De Morgan's Rules.
9. Sets of natural numbers.
10. Peano's Postulates.
11. Proof by Mathematical Induction.
12. Sets of integers.
13. Sets of rational numbers.
14. Sets of real numbers.
15. Sets of complex numbers.

Tasks for Module #2: Linear systems.

At one topic per day, this module can be completed in 3 weeks.

1. Linear systems of equations.
2. Underdetermined and overdetermined systems.
3. The process of Gaussian elimination.
4. Row vectors and scalars; addition and subtraction of vectors.
5. Scalar product.
6. Inner product.
7. Matrices, and matrix addition and subtraction.
8. Scalar product.
9. Matrix multiplication.
10. The determinant of a matrix.
11. Minors and cofactors of determinants.
12. The vector product.
13. Column vectors.
14. Expressing a system of linear equations as the product of a matrix and a vector.
15. Gaussian elimination in matrix form.

Tasks for Module #3: Relations and functions.

At one topic per day, this module can be completed in 4 weeks.

1. Relations.
2. Ordered pairs.
3. Cartesian products.
4. Domain and range.
5. Functions.
6. Variables and constants.
7. Constant functions.
8. Absolute value functions.
9. Arithmetic of functions.
10. Composition of functions.
11. Graphs of functions.
12. Intercepts.
13. Domain and range of a graph.
14. Symmetry in graphs.
15. Asymptotes.
16. Inverse functions.
17. Monotonic functions.
18. Identity function.
19. Functions derived from equations.
20. Implicit functions.

Tasks for Module #4: Algebraic functions.

At one topic per day, this module can be completed in 3 weeks.

1. Polynomial functions.
2. The fundamental theorem of algebra.
3. The remainder theorem.
4. The factor theorem.
5. The number-of-roots theorem.
6. Synthetic division.
7. Roots of polynomials.
8. The rational root theorem.
9. The upper and lower bound theorems.
10. The conjugate pair theorem.
11. Real roots of polynomials.
12. Rational functions.
13. Asymptotes.
14. Partial fraction decomposition.
15. Linear factors and quadratic factors.

Tasks for Module #5: Transcendental functions.

At one topic per day, this module can be completed in 2 weeks.

1. Exponential functions and graphs of exponential functions.
2. Properties of exponential functions.
3. Natural exponential functions.
4. Logarithmic functions.
5. Properties of logarithmic functions.
6. Common logarithms.
7. Natural logarithms.
8. Exponential equations.
9. Logarithmic equations.
10. Change of base.

R004 Trigonometry

The topic list for this project is: trigonometric functions, trigonometric identities, the solution of triangle problems, complex numbers, and vectors.

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Project Objectives:

• To master the basic ideas of trigonometric functions and their inverses.
• To be able to recognize trigonometric functions by their graphs.
• The ability to use trigonometric identities.
• The ability to solve right triangle problems.
• The law of sines.
• The law of cosines.
• Understanding the complex plane.
• To gain familiarity with the exponential form of trigonometric functions.
• The complex form of trigonometric functions.
• To become familiar with the notion of vectors.
• To become familiar with the complex form of vectors and the vector form of complex numbers.

Modules:

1. Introduction and trigonometric functions.
2. Trigonometric identities.
3. The solution of triangle problems.
4. Complex numbers.
5. Vectors.

Tasks for Module #1: Introduction and trigonometric functions.

At one topic per day, this module can be completed in 3 weeks.

1. Begin a notebook for the project. This will be worth 1 point upon completion.
2. Radians and The unit circle.
3. Sine as a ratio of triangular elements, and sine as a function on the unit circle.
4. Cosine as a ratio of triangular elements, and cosine as a function on the unit circle.
5. Tangent as a ratio of triangular elements, and tangent as a function on the unit circle.
6. Cotangent as a ratio of triangular elements, and cotangent as a function on the unit circle.
7. Secant as a ratio of triangular elements, and secant as a function on the unit circle.
8. Cosecant as a ratio of triangular elements, and cosecant as a function on the unit circle.
9. Arcsine as a ratio of triangular elements, and arcsine as a function on the unit circle.
10. Arccosine as a ratio of triangular elements, and arccosine as a function on the unit circle.
11. Arctangent as a ratio of triangular elements, and arctangent as a function on the unit circle.
12. Arccotangent as a ratio of triangular elements, and arccotangent as a function on the unit circle.
13. Arcsecant as a ratio of triangular elements, and arcsecant as a function on the unit circle.
14. Arccosecant as a ratio of triangular elements, and arccosecant as a function on the unit circle.
15. Graphs of trigonometric functions.

Tasks for Module #2: Trigonometric identities.

At one topic per day, this module can be completed in 2 weeks.

1. Reciprocal identities and quotient identities.
2. Pythagorean identities and negative-angle identities.
3. Cosine of a sum or difference.
4. Cofunction identities.
5. Sine of a sum or difference.
6. Tangent of a sum or difference.
7. Double-angle identities.
8. Half-angle identities.
9. Sine of powers.
10. Cosine or powers.

Tasks for Module #3: The solution of triangle problems.

At one topic per day, this module can be completed in 2 weeks.

1. Trigonometric functions of special angles.
2. Reference angles.
3. Right triangles.
4. Oblique triangles.
5. The law of sines.
6. SAA and ASA triangles.
7. Area of a triangle.
8. SSA triangles.
9. Law of cosines.
10. Solving SAS triangles and Heron's formula for the area of a triangle.

Tasks for Module #4: Complex numbers.

At one topic per day, this module can be completed in 3 weeks.

1. Imaginary numbers.
2. Complex numbers.
3. Complex solutions of equations.
4. Addition and subtraction.
5. Multiplication and conjugates.
6. Division.
7. The complex plane.
8. Polar form of complex numbers.
9. Trigonometric form.
10. Product theorem.
11. Quotient theorem.
12. De Moivre's theorem.
13. nth root theorem.
14. Polar coordinates.
15. Parametric equations.

Tasks for Module #5: Vectors.

At one topic per day, this module can be completed in 1 week.

1. Angles and magnitudes.
2. Vector operations and unit vectors.
3. Inner product.
4. Vector product.
5. Triple products.

About the Lessons:

The lessons of this course will be written and administered using Mathematica. You will need either MathReader or a copy of Mathematica to read these lessons. MathReader is freely available and can be found here. If you are taking this course officially you can discuss the matter with your instructor who can get you a copy of Mathematica (free for six months, or the student version for $140 for as long as you are a MAST/SAS student with the ability to upgrade to the professional version for $350, at toal of $590 — a savings of over $600).

Instructors for this Course

George Hrabovsky

Student Projects

There are no student projects at this time.

Some Useful References

Paul R. Halmos [1960], Naive Set Theory, Springer [1974].

This is one of the most engaging books that I have seen on the subject of set theory.

Bruce E. Meserve [1953], Fundamental Concepts of Algebra, Dover [1982]

An excellent book on algebra beginning with the notions of set theory and our number system.

Bruce E. Meserve [1955], Fundamental Concepts of Geometry, Dover [1983]

A companion volume to the algebra book, and every bit as good.

Richard Courant, Herbert Robbins, [1969], revised by Ian Stewart [1996], What is Mathematics?, Oxford University Press [1996]

This is a wonderful collection of mathematical chunks revealed in an engaging and understandable style.

John P. D'Angelo, Douglas B. West [2000], Mathematical Thinking, Prentice Hall.

This is a very modern treatment of logic and methods of proof. Along the way it also develops a lot of set theory, number theory, combinatorics, and analysis.

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