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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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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.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

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?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + 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 human-made 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

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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 Representation

Appendices, Volume 3, Why Slopes and More Math.

Some fractions can be written in the form


where m is a natural number and k is an integer. Such fractions have a finite decimal expansion.

Theorem: If a fraction r can be written in the form p/q where p is an integer and q is given by a products of 2s and 5s (i.e. has no other prime factors), then p/q can be written in the form


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

= 0.6666 where the 6 repeats

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

  • 1.41421
  • 1.414213
  • 1.4142135
  • 1.41421356
  • 1.414213562
  • 1.4142135623

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 Sequences

Imagine 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:

  • (Decimal Perspective): For every whole number k, there exist a whole number m such that g(p) will agree with g(q) to k decimal places when p > m and q > m
  • (Decimal Free Perspective): For every positive number E > 0, there exists a whole number m such that |g(p) - g(q)| <e if p > m and q > m.

Both conditions are equivalent. Each implies the other. Again, why depends on how you think of the real numbers.

Each infinite decimal expansion can be thought of as a Cauchy Sequence in which the k-th term gives the limit, a real number, to say k-decimal places.

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 k-decimal places because

For every whole number k, there exist a whole number m such that g(p) will agree with g(q) to k decimal places when both p and q are greater than m.

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 Decimals

The decimal-free 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 thought-based 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 thought-based 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 set-theoretic 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 set-theoretic justification of decimal expansions and their convergence of the latter. Before and after this, courses that discuss the decimal and decimal-free 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-k-1. 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 Rn

If a bounded region B is covered and partitioned by intervals whose sides have length 10-k, and an infinite S in B is given, then there must be a lexicographically least interval with infinitely many points of S inside it. As k increases, these intervals will be nested, and the lexicographical least corners of the k-th interval will yield the decimal expansion of a limit point -- approach the limit in a lexicographically increasing fashion. Exercise: Verify the details and show that writer has not made any mistakes. (In this process we could define the lim inf of a set in R^n with respect to the lexicographic ordering of points.

Here (x1, x2, ..., xn) is lexicographically > (y1, y2, ..., yn) iff there is a whole number k (1 < k < n) such that xm = ym if m < k and xk < yk.

Recap: Cauchy Sequences

  • [Play Video] 4½ minutes: Algebraic View of Limits. Example involving sums and quotients.
  • [Play Video] 5½ minutes: Limits and Error Control for Linear Expressions
  • [Play Video] 2¾ minutes: Error Control to N decimal Places, say 5 or 10.
  • [Play Video] 3¼ minutes: Limits as Error Control for an unlimited number of decimal places.

In dealing with real numbers, we assume that each finite and infinite decimal expansion defines a real number. When two numbers differ by [1/2] ·10-k > 0, their decimal expansions are said to agree to k decimal places. Convergence of a sequence to a limit L can now be expressed in terms of decimal numbers or significant digits: For any whole number k, there is a whole number N, such that all terms in the sequence after the first N agree with the limit L to k decimal places.

Convergence here corresponds to the ability in principle, if not in practice, to patiently compute a decimal or binary expansion to an unlimited number of places.

Error control in practice requires a rate of convergence estimate to say how large N must be to obtain k decimal places. We may distinguish between convergence arguments which says there is always N and convergence arguments which give N as an easily-computed function of k - convergence in principle versus the desired situation in which the rate of convergence can be described and computed.

A Cauchy sequence f(n) has the following property: For each whole number k, there is a whole number N with the following property: all terms in the sequence after the first N-1 agree with each other to at least k decimal places. This property allows us to define and compute in principle an infinite decimal expansion. This expansion is assumed to define a unique real number: the limit L of the Cauchy sequence.

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]

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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 head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based 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 story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - 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; pattern-based 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 cross-products, 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 how-tos 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.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

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.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.

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