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geometric development
Coordinates, relative or absolute?
Coordinates may be given relative to a choice of unit
length and direction (a unit vector) along the coordinate axes of a map.
Or, equivalently, coordinates may be given relative to a choice of unit
length and a choice of positive direction for the coordinate axes. In both
cases, these relative coordinates are ordered pairs of signed numbers.
Coordinates may also be given absolutely relative to a
choice of unit length and choice of positive direction for each coordinate
axis. For example, a point in a planar map may be determined by absolute
coordinates
[+5 cm, -6
cm]
where here the unit of length is the centimeter cm.
Implicit here (a first example) is the multiplication of the unit length
by a signed number. Implicit here (another first example) is a
multiplication of the unit and unit vectors along the coordinate axes by
signed numbers.
Addition of vectors (displacements) in the plane and
more specifically collinear vectors in a line, their multiplication by
signed numbers (coordinates) and their representation as signed numbers
multiplies of a unit vector implies is or consistent with definition of
the addition and multiplication of the signed number multipliers alone,
apart from their role in representing collinear vectors as multiples of a
given unit vector.
Advanced Theory
- Unsigned
Reals Numbers - use of unsigned decimals as coordinates.
- Signed
Coordinates - Introduction of real numbers by prefixing signs to
hitherto unsigned numbers.
- Plane
Vectors - Navigation - use of arrows or vectors in describing
piecewise linear paths in the plane; Head-to-tail addition;
Associativity of in place head-to-tail addition.
- Horizontal
Vectors & Adding
Vector Multiples of unit vectors]. Addition of horizontal, more
generally collinear, vectors that represent displacements, AND
properties of this addition - commutativity included.
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Adding
Signed Numbers. The addition of signed numbers A and B is defined
so the addition of multiples A and B of a vector equals
the multiple A+B the vector.
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Multiplying
Signed Numbers. The product or multiplication of signed numbers is
defined so the multiplication by signed number A of a signed number
multiple B of a vector is equals the multiple AB of that vector.
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Distributive
Law for Reals. The sum of collinear vectors given by multiples A
and B of a nonzero k should not change if k = c m where m is another
vector. Two methods of expressing the sum as multiple of c lead to the
distributive property (A+B)C = AC + BC for signed real numbers.
-
[Real
Numbers Axioms] The foregoing considerations imply a superset of
the real number axioms assumed in modern mathematics curricula (or
derived in a context free manner in pure mathematics.)
- Modular
or Remainder Arithmetic for real numbers- Here is real number
generalization of modular or remainder arithmetic for whole numbers.
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Real or Signed Numbers may be introduced as Coordinates
with signs
in 2D and then 1D
(Using maps & coordinates to introduce rectangular coordinates)
Unsigned numbers and absolute or relative lengths may be used in ordered
pairs to locate points on rectangular map, with say the origin at a bottom-left
corner. If that small map is extended or placed on a larger rectangular map in
which the origin of the small map is not at the bottom, left corner of the
larger one, the small map ordered pair, coordinate system may be extended
through the use of negative numbers written in the manner -5
with negative signs in the super-prescript before a whole number 5. Here
positive numbers, for example 6, may be employed in place of and used like
unsigned numbers.
Issue: The use of signs + and - in the super-prescript position
before whole numbers and decimals appeared in my mathematics education, but
was not used in practice with unsigned fractions (a/b) to generate signed
fractions. With the latter, signs were employed in prefix position and not in
prefix, superscript position). The superscript placement of signs, positive
and negative, prefixes appears to be optional. It stems from or before the
Modern Mathematics curricula of the mid-1950s.
Arithmetic with Signed (Real) Numbers
See how to develop an operational command of arithmetic signed numbers
(integers, rational, decimals and in general). This may be met or illustrated
first with integers, then rational numbers and decimals. Here decimal
arithmetic is done exactly or approximately
Additive Inverse - Negative of a Number A:
For A = sign(A) length (A) is nonzero, the negative of A is -A = co-sign(A)
length(A) = the additive inverse of A. If A is 0, the negative of A is 0 and
additive inverse of A is zero. (Saying how to calculate A defines it.)
Addition of Signed Numbers:
The sum of two signed (a.k.a real) numbers A and B is given as follows
- If A and B have the same sign then
A+B = (common sign)( Magnitude(A) + Magnitude(B))
= (common sign)(sum of the addend's magnitudes)
Here the magnitudes are unsigned real numbers given by decimal or fractions
etc.
- If A and B have opposite signs and are equal in magnitude (length) then A
and B
are additive inverses with B = -A and -A = B, and
A+B = 0
- If A and B have opposite signs and unequal in magnitude (length) then
A+ B = (sign of Biggest)( Biggest - Smallest)
= (sign of longest) (Longest - Shortest)
If sign(A) is + or +1 then co-sign(A) is - or -1. And if sign(A) is - or -1
then co-sign(A) is + or +1.
Subtraction
The rule B - A = B + (-A) allows all subtractions of a signed number A to
be expressed (rewritten) as additions involving the negative inverse of A.
Product of Signs:
(+)(+) = +
(+)(-) = -
(-)(+) = -
(-)(-) = +
Multiplication of Signed Numbers:
Next if A and B are signed numbers, their product
AB = (sign A)(sign B) [(length of A)] [(Length of B)]
= [(sign A)(sign B)] [(magnitude of A)(magnitude of B)]
Call this the multiply the signs, multiply the lengths
for multiplication pf pairs of signed numbers. Take the product AB to be zero if
A or B is zero.
Remark: If we define the multiplication of arrows V
by signed numbers A as follows
| A times V = { |
the vector of length (length A)(Length V)
with the same direction as V if A is positive and the opposite
direction if A is negative. |
The the law of signs for multiplication is consistent with the
associativity of this "scalar" or signed real number multiplication of
arrows:
A times (B V) = (AB) times V
whenever A and B are signed numbers and V is an arrows.
This property illuminates the law of signs and provides a geometric motivation
for it. The definition of products of signed numbers could also be
presented after the definition of products of signed numbers and arrows
was defined. There-in lies a longer route, but one that might appeal
to some as more natural.
Leading Questions: How many times does the first
arrow go into the second collinear arrow?
Division of Signed Numbers:
| Example |
First Arrow |
Second Arrow |
No of Times |
| A |
= = => |
= = = = = => |
2 or +2 |
| B |
<= = = |
= = = = = => |
-2 |
| C |
= = > |
<= = = = = = |
-3 |
| D |
<= = |
< = = = = = = |
3 or +3 |
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Leading Question: How many times does a
arrow of length q divide or go into a arrow of length p when (i) they
are collinear with the same direction; and (ii) they are collinear with
the opposition directions.
Answer for (i) is p/q
Answer for (ii) is - (p/q).
Now each signed number may be identified with an
collinear arrow in the positive or negative direction of a coordinate
axis.
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Example
Revisited |
First Arrow
or Number |
Second Arrow
or Number |
No of Times |
| A |
= = => (+3) |
= = = = = => (+6) |
2 or +2 |
| B |
<= = = (-3) |
= = = = = => (+6) |
-2 |
| C |
= = > (+ 2) |
<= = = = = = (-6) |
-3 |
| D |
<= = (-2) |
< = = = = = = (-6) |
3 or +3 |
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In general, if A and B are signed numbers, their
quotient
A/B = (sign A)(sign B) [(length of A)/(Length of B)]
= [(sign A)(sign B)] [(magnitude of A)/(magnitude of B)]
Check that in revisited examples A to B above. That
suggest the following rule or convention for the division of signs.
(+)/(+) = (+)(+) = +
(+)/(-) = (+)(-) = -
(-)/(+) = (-)(+) = -
(-)/(-) = (-)(-) = +
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Multiplicative Inverse (Reciprocal):
If A is nonzero, then the multiplicative inverse (a.k.a
reciprocal) of A is
|
A-1 =
|
1
A
|
=
|
sign(a) |
.
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1
length(A)
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Division
The rule B/A = B (1/A) allows division involving a signed number A to be
expressed (rewritten) as products involving the multiplicative inverse of A.
Comparisons of Signed Numbers:
Greater in Magnitude Comparison:
The magnitude (or length) of the signed Numbers -10, +5,
-1, 0, +3 can be compared. We see that -10 has the largest magnitude, namely
10, while 0 has the small magnitude and that is 0. Here -10 is greater
in magnitude than say +5 while 5 is greater than + 3 in magnitude.
Less Than Comparison and the LESS THAN sign <
Examples:
-
Observe 15 = 10 + 5 or 10 = 15 -5. Here 10 is 5 less
than 15. We say 10 is 5 LESS THAN 10,
and write 10 < 15 (by 5)
-
Observe 2 = -4 + 6 or -4 = 2 -6. Here -4
is 6 LESS than -2, and we write -4 < 2 (by 6)
-
Observe -8 = -15 + 7, or -15 = -8 - 7. So -15 is
7 less than -8, and we write -15 < -8 (by 7)
The by N part in parentheses gives the difference.
The part is optional.
Definition (Algebraic Form): a first signed
number A is less than a second signed number B and we write A < B by
when A = B - C for some positive number C
More Than Comparison and the MORE THAN sign >
Examples
-
Observe 15 = 10 + 5. Here 15 is 5 more than 10. We say
15 is 5 more than 10,
and write 15 > 10
-
Observe 2 = -4 + 6. Here 2 is 6 more than -4,
and we write 2 > -4
-
Observe -8 = -15 + 7. So -8 is 7 more than -15, and write -8
> -15
Definition (Algebraic Form):In general a first number A
is more than a second number B and we write A > B when the first number
A is given by the second number B plus a positive number C. That is,
when A = B + C exceeds B by a positive number C.
Remark (Name Change Suggestion): Instead of calling the
sign >, the greater than sign, teachers and students should
call it the more than sign. That may help because primary and junior
high school students learn to compare unsigned number by magnitude and not by
the more positive idea. The name change is consistent with calling the sign
<, the less than sign. See below. (The webpage Reference: Rename
the Greater Than Sign written earlier suggests calling > the more
positive sign instead of greater than sign. However the phrases (i)
-10 is +4 more positive than -14 and (ii) -10 is greater than -14 are as
appealing to my ear as the phrase -10 is +4 more than -14.
To Do: Add or link to a lesson explaining how to use the more than or
more positive than concept to manipulate inequalities - to obtain properties
of inequalities - how they are preserved or reversed under addition of terms
and multiplication by signed numbers.
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Odds & Ends
Group I
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
8. Complex No. Applet
7. Trig and Complex No.
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24. The 20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Biology - Growth & Decay
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More
Group II
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
Two Gaps
[ Back ] [ Up ] [ Next ]
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For
Senior
High School & Calculus Students
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<| (o) (o)
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-/[]\-
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Words to clearly
introduce algebra and variables
have been missing in course design. For people who cannot do
algebra,
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the missing words may
explain or ease their difficulties. Volume 2 ,Three
Skills for Algebra, in Chapters
8 to 14 & 18 etc, puts words before symbols to
providing the missing words in a way that enrich the
comprehension of all. Those words form the middle part of a algebra
(and logic) lessons aimed at helping or improving all
of high school mathematics and also calculus course
design & delivery.
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For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide. If your
calculus questions is not answered here, submit
it. Over time, that may complete the site development of
calculus.
For Parents: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly maths
skill development booklets for ages 4-14.
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Mostly
For High
School
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students. Question for Tutors: When do
you use and when you skip the stick diagram method
here?
Fraction
Skills, thought-based development, Ages 10 to 14 may need a
tutor. Students who have to understand in order
to do may like the development in all or part.
For Senior
High School Mathematics & Calculus
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
Tutors & Instructors:
These lessons introduce skills differently Would you
recommend them?
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More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
for- & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- Free
Advice and Directions for teaching primary & high school maths
will be given in online meeting place with voice &
whiteboard.
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students. Offering ideas to change
education makes this site different. Nothing
ventured, nothing gained. Site material is
mathematically correct, and where not, please report
errors. The two level program POMME in the site
entrance implies multiple paths for instruction. Supporting
those paths in turn implies a clear destination for
site development and perhaps a new name.
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