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Real Numbers, Decimal Representation
Some fractions can be written in the form
m
10k
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
m
10k
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
2
3 |
= |
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.
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Calculus Guide
Section Entrance
Real Player Videos
My First Steps
About Calculus
1. Regular First Steps
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material
Up
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
Up
Reference Material: - Light
reading for calculus.
Vol 2, Three
Skills for Algebra covers many topics in algebra and logic
that students starting calculus should have mastered or will have to
master sooner or later. Also includes arithmetic review problems to
catch common mistakes.
Vol. 3, Why
Slopes & More Maths, gives starter lessons for differential
and integral calculus.
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volumes 2 and 3 before calculus and during it. |
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For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
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.
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Senior
High School &
Calculus Students
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Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
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For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
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)
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Many 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
Use Forward & 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.
More For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- POMME, a two
level program for instruction K1-14
- 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.
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).
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Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
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