Number Theory

Number Theory & Practices

- What Teachers & Tutors Need to Know -

A. Start of Number Theory/Practices

The webpages

starts with a development of enumeration principles and consequences for whole and natural numbers. The last two items complement the site development of Fractions - the latter link is more extensive.

The student specializing in pure mathematics may recast all or most in terms of the set-theoretic development of modern mathematics (1905 onward) and provide missing details using mathematical induction.

Primary School Teachers: Read items A, C and D (the easy part) to see what primary school students will need in high school if not before. See too the site development of 2. Integers - Intro to Signed No.s 3. Fractions - fully explained. 4. Fractions with Units 6. Solving Linear Equations. For most students, the thought-based development of skills and concepts can be skipped, or better yet, be partially given in a way that does not overwhelm students. The latter will be needed where a student balks at accepting arithmetic methods without. A knowledge of the skills your students will have to master soon or later may improve your lesson planning and delivery. This advice may fill gaps left by teacher training programs, especially those that did not require mathematics skill mastery.

Junior High School Teachers: Senior high school and college mathematics requires arithmetic skills with decimals, fractions, primes, percentages and signed numbers in a reliable, full-strength manner. You job is to provide that standing if you can on the results of primary school instruction. Give yourself a firmer base for that by understanding to the greatest extent possible items A to D below, and optionally Item E. See too the site development of 2. Integers - Intro to Signed No.s 3. Fractions - fully explained. 4. Fractions with Units. These further algebra & logic lessons (Solving Linear Equations included) will also help in high school mathematics, all years. As said to primary school teachers, knowledge of the skills your students will have to master soon or later may improve your lesson planning and delivery. This advice will fill gaps left by teacher training programs if you did or did not specialize in secondary mathematics instruction. The site coverage of primes and their employment.(see below) and the algebra & logic lessons gives methods to make the hard easier, methods almost surely not previously taught in mathematics teacher training programs.

Senior High School Teachers: Follow the directions for junior high school teachers and be sure to read item E and all the algebra & logic lessons,.

A. Decimal Place Value Revisited.

Decimal Place Value
Place Value Reinforcement
Addition Method
Comparison Method
Subtraction Methods
Multiplication Methods
Division Methods
Long Division Continued

Here are explanations for decimal place value methods for comparing, adding, subtracting, multiplying and dividing whole numbers. In particular, The first six explore decimal place value and how with the aid of the distributive law, it justifies methods for comparison, addition, subtraction, multiplication and division of whole numbers. Full explanation of column methods for arithmetic are included here.

See too (i) New (flash based) webvideos on Decimal Arithmetic & Integers (ii) Arithmetic HOW-TO and (iii) older (real-player) Arithmetic Videos for more

B. Remainder Arithmetic I

The lesson Remainder Arithmetic explores and justifies the decimal-based rules for recognizing multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. Consideration of arithmetic modulo these whole numbers lead to remainder arithmetic, computing methods for the remainders when whole numbers are divided by 2, 3, 4, 5, 6, 7, 8, 9, 10 or 11. Divisibility rules are included by the special cases where the remainder is zero. Remainder calculations here for the number 7 may be unique.

C. Primes Numbers

Primes & Composites
Primes Factorization Theorem
GCMs and LCMs from Primes
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
N-th Roots and Primes

Webpages Primes & Composites, Prime Factorization, Arithmetic Videos, Square Roots develop an enriched computational view of primes and factors for use in arithmetic with whole numbers and fractions and in further mathematics subjects at the high school and college level.

D. Setting the stage for Real Numbers
- the easy part

Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring

- More

Arithmetic With Infinite Decimals Expansion s
Ratio of Simple Fractions
Ratio of Decimal Fractions

The Pythagorean Theorem geometrically gives lengths that are square roots of 2, 3, 5 and so on (here further primes) multiples of another side or unit length. But the hope that these square roots are given by proper or improper fractions (ratios of whole numbers) does not agree with what has come before in number theory. So for the sake of consistency, to avoid disagreement, the theory of numbers may only proceed by accepting that these square roots cannot be given by improper fractions.

E. Remainder Arithmetic for Real Numbers (Remainder Arithmetic II in navigation bars)

This pages extends the modular or remainder arithmetic developed for natural numbers - Explore concepts useful for the discussion of sinusoidal functions.

F. Thought-Based Account of Real Numbers (advanced stuff)

- the easy part
Unsigned Reals Numbers
Signed Coordinates

- the rest (for very few)
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals

H. Real Numbers Axioms -These algebraically describe the arithmetic properties of real numbers.

Chapter 18 uses words and examples to make the algebraic description more accessible.
But there is another option. Accountants obtain sums of positive and negative amounts by adding subtotals where the subtotals come from a partition or grouping of the amounts in disjoint, that is non-overlapping sets. And in learning to count, children also shown how to partition the objects being counted into groups and then add the sub- counts. The mathematical justification of these practice starting from say from given assumption of commutative and associative laws for addition is long and beyond the reach of most students, and not given and so not explicitly sanctioned in mathematics courses. The further practice of partitioning or grouping terms in a product to express the latter as a product of subproducts also appears in the prime number factorization and its implications for operations on fractions, roots and monomials where the terms of the monomials may be numbers, letters or symbols denoting numbers, or units of measure. The practices of partitioning into groups or subsets for the sake of counting using subcounts, totaling using subtotals and multiplying using subproducts are present in the mastery of decimals, fractions (decimal or not), polynomials and weights and measures. Describing these practices and giving some or all as assumptions provides an axiomatic base to sanction and make clearer common practices in school mathematics and in its applications. The sanction may involve more words then symbols to describe the practices in courses before university programs specializing in pure mathematics derive the fuller axioms. The immediate end here is to justify those practices and make them and their consequences easier to recognize and less complicated to see and master. For the latter option, seen August 2010, material to show how may be implemented in class remains to be composed.

H. Thought-Based Account of Real Numbers (advanced stuff)

- the easy part
Unsigned Reals Numbers
Signed Coordinates

- the rest (for very few)
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals

These pages offer a thought-based framework for a concrete derivation of the field properties of real numbers. Familiarity with fractions is assumed.

The remaining pages provide a logical development of the properties of real numbers and real number arithmetic from enumeration principles, assumptions about vectors in the line and plane, properties of fractions and decimal representation of unsigned fractions and unsigned real numbers to the field properties of real numbers. Some unique arguments appear here.

The proof here of the distributive law for real numbers is based on the notions that the real number line is a one-dimensional vector space over the set of real number and the choice of a base vector, the unit vector, should not affect the computations. Indeed, the latter principle,, that is the choice of a base vector for vector addition should not affect the result of vector addition, is a consequence of the distributive law. A similar proof is indicated for obtaining the distributive law for complex numbers. Details may be given later.

Remark for Studies in Pure Mathematics: (I) The assumption of set theory practices (axioms) by a long deductive path , too long for most students, may give undergraduate students specializing in mathematics, a thought-based development of axioms for real numbers without explicitly sanctioning nor mentioning the common arithmetic skills. But most who see those axioms for real numbers will not see the derivation. But (II), a thought-based development based on counting and geometric practices (principles may be a too strong a word) with whole numbers, fractions and decimals may quickly imply those axioms while extending and endorsing the common figuring skills. Option (I) is more inclusive and it serve as a prequel to option (II) in a manner that would enrich and complete the comprehension of students specializing in mathematics. The latter provides a context for the Start of Number Theory/Practices above

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Number Theory & Practices

A. Start of Number Theory/Practices

B. More Number Theory & Practices

Number Theory & Practices

Number Theory