Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. Early High School Algebra
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? Early High School GeometryMaps + Plans Use  Measurement use maps, plans and diagrams drawn to scale.  Coordinates  Use them not only for locating points but also for rotating and translating in the plane.  What is Similarity  another view of using maps, plans and diagrams drawn to scale in the plane and space. Many humanmade objects are similar by design.  7 Complex Numbers Appetizer. What is or where is the square root of 1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of 1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.  Geometric Notions with Ruler & Compass Constructions : 1 Initial Concepts & Terms 2 Angle, Vertex & Side Correspondence in Triangles 3 Triangle Isometry/Congruence 4 Side Side Side Method 5 Side Angle Side Method 6 Angle Bisection 7 Angle Side Angle Method 8 Isoceles Triangles 9 Line Segment Bisection 10 From point to line, Drop Perpendicular 11 How Side Side Side Fails 12 How Side Angle Side Fails 13 How Angle Side Angle Fails 
whyslopes.com >> Advanced Calculus  Volume 3 Appendices >> PostScript  For and Against Decimal Perspectives Next: [B1 Pigeon Hole Principles from combinatorics.] Previous: [A1.Introduction.] [1] [2] [3][4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] Real Numbers, Decimal RepresentationAppendices, Volume 3, Why Slopes and More Math. Some fractions can be written in the form
m where m is a natural number and k is an integer. Such fractions have a finite decimal expansion.
m Now fractions with denominators with prime factors other than 2 and 5 do not have finite decimal expansions. They have periodic decimal expansions. For example
Here the infinite decimal expansion may be found by long division. Long division is done until the expansion starts to repeat. Arithmetic with fractions can be done directly and exact without decimal expansions, or approximately with decimal expansion. In approximate calculations, only finitely many decimals are used  the more, the better, for the sake of accuracy. It can be shown that arithmetic with periodic decimal expansions produces results with periodic decimal expansions. Error control with approximate arithmetic depends on the continuity or error control analysis of addition, subtraction, division and multiplication. The whole number 1 = 1.000 and the repeating decimal expansion 0.99999 give two decimal representations of the same number. The first expansion 1 = 1.000 (finitely many zeroes or none) is finite and exact. The second decimal expansion 0.9999 (9 recurring) represents a sequence of fractions 0.9, 0.99, 0.999, 0.9999, whose limit equals 1. When a number has a finite and an infinite decimal expansion, the finite one is simpler to use, but both are valid. The square root of 2 is not a fraction. But there is a sequence of decimal numbers
whose squares have the limiting value 2. The error (difference between) the limit 2 and the square decreases as more and more decimal places are used. On a coordinate line, any line segment whose length can be approximated by an infinite decimal expansion is considered to be a real number. Here continuity or error control arguments allow us to do arithmetic with infinite decimal expansion and compute the results with unlimited error control to an unlimited number of places. We assume that each finite and each infinite decimal expansion gives us a real number. Cauchy SequencesImagine we have an infinite sequence of numbers g(1), g(2), g(3), ... This sequence is said to be a Cauchy sequence if the one of the following properties holds:
Both conditions are equivalent. Each implies the other. Again, why depends on how you think of the real numbers.
Now every Cauchy Sequence has a limit L. To show this, we assume that specifying in principle how to compute the decimal expansion of L determines the value of L. (The number pi = 3.14... is an example of real number that can be computed to million of decimal places. The number pi is given by the limit of this decimal expansion.) Now if we have a Cauchy sequence g(1), g(2), g(3), ... , how do we determine the first k decimal places of a limit L. The answer is simple. According to the decimal perspective we may compute L to kdecimal places because
So given k, we may choose or find in principle, a whole number m with the property that g(p) and g(q) will agree to k decimal places whenever both p and q are more positive than m. Take the decimal expansion of g(m+1) to k decimal places. This decimal expansion to k places tell us how to compute L to k decimal places. Since k can be as large as we like, that is, arbitrary, we can in principle determine every digit in the decimal expansion of a number L. Simply go far enough along the sequence. By this construction, a limit L of the Cauchy sequence g(1), g(2), g(3) can in principle be computed. That is enough to say the limit L exist at least in principle. The argument using decimal free perspectives of real numbers is more complicated. The Role of DecimalsThe decimalfree set theoretic view of mathematics reached it almost final form in the 1920s. It took another 30 years, that is, until the 1950s, for the set theoretic view of mathematics to be adopted in mathematics departments. The modern mathematics movement in the 1960s was intended to spread or provide a setting for the teaching of the set theoretic perspective. The set theoretic perspective began about the mid 1800s, and it was used in the period 1900 1930 to provide a strict thoughtbased foundation for computations  the arithmetic based part of mathematics  a foundation (hopefully) free of contradictions and inconsistencies. This set theoretic perspective was not developed for ease of exposition. The initial aim in studying sets was not to provide a foundation for arithmetic based mathematics. In the set theoretic approach to mathematics after arithmetic (counting included), the decimal perspective of real numbers was not necessary. So it was put aside. In contrast, the common knowledge of mathematics is based on counting, a decimal knowledge of arithmetic and real numbers, and the use of simple formulas. This common knowledge is introduced and hopefully explained in elementary school in a thoughtbased manner. The common knowledge presently encompasses counting, arithmetic and the use of simple formulas. The decimal expansion of real numbers provides a concrete sense of convergence. Unfortunately, in the zeal to derive the set theoretic perspective from first settheoretic principles or assumptions about real numbers in our high schools and colleges, the decimal perspective was put aside at least partially. That is, while the decimal representation of whole numbers and real numbers was employed in computational examples in algebra, trig, chemistry, physic, business and calculus, the chains of reasoning emphasized in algebra and calculus typically made no mention of decimals (nor units). Decimals (and sometimes units) were used in many computational subjects yet not recognized nor sanctioned in math courses axioms. Courses on analysis (advanced calculus) could be made more accessible to students by detailing in them a settheoretic justification of decimal expansions and their convergence of the latter. Before and after this, courses that discuss the decimal and decimalfree perspective would be agreeable both to students of analysis and students who just assume the convergence of decimal expansion. Ease of exposition is the motivation for this suggestion. Remark: A mathematics or science student could follow the more accessible decimal perspective in a calculus and then in a later analysis (or advance calculus) course, meet the set theoretic perspective justification and/or reformulation of the decimal arguments. Does rigor in haste lead to rigor mortis? Remark (for advance students): Appendices in Volume 3, Why Slopes and More Math, provide the decimal and decimal free perspective of the basic theorems in calculus. For instance, the Bolzano Weierstrass Theorem that every infinite set in a closed interval has a limit point can be viewed as consequence of the Pigeon hole principle. The leftmost limit point has a decimal expansion computed to k decimals by covering the interval with nicely aligned subintervals of length 10**(k) and locating the leftmost one with infinitely points (or concluding that such an interval most exist. The latter must contain the leftmost interval of length 10^{k1}. Thus a sequence of nested intervals with left end points converging to the "lim inf" (technical expression) of the set is obtained. Remark (also for advanced students): Lexicographic ordering of points and intervals with sides of length 10^{k} should extend this argument to bounded infinite sets or sequences in R^{n}
Here (x_{1}, x_{2}, ..., x_{n}) is lexicographically > (y_{1}, y_{2}, ..., y_{n}) iff there is a whole number k (1 < k < n) such that x_{m} = y_{m} if m < k and x_{k} < y_{k}.
whyslopes.com >> Advanced Calculus  Volume 3 Appendices >> PostScript  For and Against Decimal Perspectives Next: [B1 Pigeon Hole Principles from combinatorics.] Previous: [A1.Introduction.] [1] [2] [3][4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] 
Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic? Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based ReasonOnline Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge. Site Reviews1996  Magellan, the McKinley Internet Directory:Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it. 2000  Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; patternbased reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001  Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot and
crossproducts, the cosine law,a converse to the Pythagorean Theorem
2002  NSDL Scout Report for Mathematics, Engineering, Technology  Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and howtos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005  The NSDL Scout Report for Mathematics Engineering and Technology  Volume 4, Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Senior High School Geometry

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions. Calculus Starter Lessons
Why study slopes  this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. 