5 Integers
1 Integers as Coordinates
2 Integers Multiplies of a Unit Moverment
3 Adding Movements with same direction
4 Adding Movements wiht opposite directions
5 Zero Movement and Additive Inverses
6 Multiplication by Natural Numbers
7 Multiplication by Signs
8 Multiplication by Signed Numbers - Integers
9 Multiplying Integers
10 Integer Multiplication Formulas
11 Adding Integers - Formulas and Examples
12 Adding Integers - More Examples
13 Subtraction with Additive Inverse
A Associative Law - Theorectical Note
B Integer Long Division - Multiple Choices
C Divisibility by 11 - Integer Recognition Method
D Remainders Modulo 11 Pair Rule
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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