7. Arithmetic with Infinite Decimal Expansions
Operational Viewpoint:
In brief, arithmetic involving numbers given or represented by infinite
decimal expansions is done by approximation with the hope or assumption
(justified by error control, continuity and convergence analysis in
calculus or beyond) that better and better results required more and more
decimals in the calculation. In practice, apart from theory and
with the aid of calculators, the number of significant or exact
decimal places in a result that depends on approximations is estimated
from which digits are changed as approximations to operands in the
calculation get more precise - involve more decimal
places.
Calculation Theory
The addition, subtraction, multiplication and division of finite and
infinite decimal expansion may be defined by a sequence of
approximations: do the calculations with say n leading decimals
from the expansion involved to get an approximation f(n). By continuity
or error control analysis, we may show that the approximations f(n) form
a Cauchy sequence and therefore define the result, another finite or
infinite expansion that gives the result. Details may be found in
the first chapter in the book Calculus by L. Bers (Holt,
Rinehart and Winston 1969, SBN 03-065240-5). The chapter in question
provides further background information on the decimal and decimal-free
representation of real numbers. Or see
Error Control Inequalities in the Advanced calculus, Volume 3 appendices
area of this site.
In practice, we do not do the error control analysis (too much work for
hand computation) and instead take as many decimal places as a calculator
permits or we examine the effect on the result (our approximation to it)
of taking fewer and then more and more decimal places to approximate the
numbers (infinite decimal expansions) involved.
The extension of our concepts of what is a number from whole numbers and
fractions to include infinite decimal expansions allows us to represent
square roots of prime number, square root of 2 included, and the
number p used in circle based
calculations - perimeter, area, trig functions.
By continuity or error control analysis, we may show that addition and
multiplication of unsigned reals are each commutative and associative;
that multiplication distributes over addition (so the distributive
property holds); that division by an unsigned number N is
equivalent to multiplication by the quotient
1
N
The finite or infinite decimal expansion of the latter can be computed
by long division continued to finitely or infinitely many places
after the decimal point.
L. Bers in his Calculus book (Holt, Rinehart
and Winston 1969, SBN 03-065240-5).mentioned above points out that the
decimal representation of real numbers is sufficient or should be
sufficient for most students in mathematics or a quantitative
discipline. The modern mathematics curricula of the 1950's depended on
decimals for representing whole numbers, fractions and irrationals but
did not discuss or explicitly sanction this nor decimal-based
arithmetic nor the discussion of limits, continuity and error control
in calculations in an unwise adherence to the decimal free nature of
pure mathematics, or its axiomatic development and codification in
terms of sets and set theory. Those ommissions left gaps in the
education of students and instructors from the late 1950's onward
where-ever modern mathematics curricula were used. The United Kingdom
(Britain) may be an exception.
Infinite Decimal Expansion in Compound Fractions
A ratio of unsigned reals is a compound fraction
M
N
in which the numerator and denominator are both unsigned real numbers.
When M and N are unsigned real numbers, then the compound fraction
M
N
can be approximated using say k decimals in the decimal expansions of M
and N. Better approximations can be found by taking k larger and
larger. Some error control analysis is required to see how accurate
the approximations will be. The equality
which holds for simple fractions and hence decimal approximations to M
and N also holds via error control arguments for M and N.
Exercises
When will the sum of two infinite decimal expansions with periods m
and n respectively (so they represent fractions in reduced form with
denominators include prime factors other than 2 and 5) be an infinite
decimal expansion with period p = l.c.m (m,n) = least common multiple of
m and n?
What happens if we add
_
10-q ×0.9999 (9 recurring, period m
=1)
to a number with an infinite decimal expansions with period n
What happens if subtract (10-q ) and then add
_
10-q ×0.9999 (9 recurring, period m
=1)
to a number with a finite decimal expansion (n =0), to a number with an
infinite decimal expansions with period n > 0 (fraction case); and
to to a number with an infinite decimal expansions, non repeating (the
irrational number case)
|
|
Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
|
|