Mathematics and Logic - Skill and Concept Development

Notes

These lessons and three appendices include exercises to consolidate and extend understanding of integers.

The lessons provide a hands-on, thought based development. Lessons assign three geometric roles to integers to develop and explain rules for integer arithmetic.

• Role I: Integer are first introduced as coordinates for points on a line, where adjacent points are a unit distance apart.
• Role II. Integers then serve as multipliers in the definition of integer multiples of a unit movement, integer multiples that can be added and multiplied by whole numbers and then integers.
• Role III. Integers themselves may describe movements, how many steps to the left or right, along a straight line, and so can be identified with movement, integer multiples of a unit movement, now called a step. That third role or identification leads allows integers to be added and multiplied.

Exercises are included in most lessons.

Three appendices cover an associative law, division quotient and remainder options for integers, and how remainder arithmetic explains the alternating sum of digits test for divisibility of decimals by 11.

Extension: This three role, geometric development and explanation of integers starting with unsigned whole numbers provides an example to follow for the development and explanation of (i) rational numbers starting with unsigned fractions; and (ii) real numbers starting from unsigned (positive) real numbers or their decimal representation. The site exposition of complex numbers continues the geometric development of number theory.

The Main Lessons

Saying how to do an operation defines it.

• Lesson 1: 1 introduces the first role of integers a signed whole or natural numbers serving as coordinates for points a unit apart along an infinite straight line.
• Lesson 2 introduces the second role of integers as multipliers in providing integer multiples of a movement, a unit movement along a line.
• Lesson 3 shows how to add integer multiples of a unit movement when those multiplies have the same (like) direction.
• Lesson 4 shows how to add opposite and same direction integer multiples of a unit movement.
• Lesson 5 introduces the idea of additive inverses for integer multiples of a unit movement. That answer the question what movement added to another will result in a zero net movement.
• Lesson 6 says how to multiple integer multiples of a unit movement by a natural number, zero or a whole number.
• Lesson 7 shows how to apply the positive and negative signs + and - to a vector. The application of the plus sign + to a movement is the identify operation while the application of the negative sign - to a movement yields its additive inverse, that is, a movement with the same direction and opposite direction.
• Lesson 8 shows how to multiply an integer multiple of a unit movement by an integer. Since each nonzero integer may be written as a plus or minus sign prefixed to a whole number, we take multiplication by a nonzero integer to be the operation of multiplying by its whole number part (also known as a length or magnitude) followed by an application of its sign to the latter product. There is a further refinement or extension of the second role of integers as multipliers of movements. The first role is to provide signed coordinates for points a unit distance apart along a line.
• Lesson 9 identifies integers with movements - so many steps in one direction or another. There-in lies a third role for integers - a movement role. That leads to an extension of lesson 8 in which multiplying integer multiples of a unit movement turns into multiplying integers by integers.
• Lesson 10 continues lesson 8 and 9 to obtain rules for products of signs and products of integers that be easily applied and learnt by rote, or seen as a consequence of lessons 1 to 9.
• Lesson 11 applies the lessons 3, 4 and 5 on the addition of integer multiples of a unit movement to the third movement role of integers. That leads to methods for addition of integers.
• Lesson 12 echoes lesson 11 in providing rules for adding integers, rules that are easily understood and repeated, with or without explanation of why they work.

The Appendices

• Appendix A (optional reading) continues lesson 11in an algebraic manner (a departure from the presentation in the other lesson) states an associative law for multiplication of integer multiples of unit movements by integers. That may - proof to be written - imply an associative law for integers.
• Appendix B (more optional reading) looks at the long division method for pairs of whole numbers for pairs of nonzero integers - long division of their whole number parts (lengths, magnitudes) implies several dividend = quotient times divisor plus remainder relations, where the sign of the remainder may be like or unlike the sign of the dividend and/or divisor.
• Appendix C links to the alternating sum of digit test for divisibility by 11 to remainder arithmetic, modulo 11, with the aid of integers. The link here is given without proof. Indeed the whole treatment of remainder calculations provides rules to apply and follow to repeatable and reproducible results, without full (algebraic) explanation of why these practices work. The explanation why is or or is to be included in the site coverage of algebra.

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