Real lengths and numbers?
An informal or vulgar account of a number line
We assume each point on a half-number line corresponds to a number:
whole, proper fraction, improper fraction or mixed number, and further
numbers.
Measurement Practice Each finite and infinite decimal expansion
gives the length of a line segment with one end at the origin of a
half-line is assumbed to a multiple of a unit length.
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No Decimal Places. Here numbers like 1, 2, 3, 4 and 100
identify a point which a whole number of unit lengths from the origin
of the half line
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Finite decimal expansions. The decimal fraction 2.4 shown
below corresponds to a point that is 2 and 4 tenths of a unit from
the origin 0.
In principle, we can a point that 5.1234 units from the origin.
Fractions have finite decimal expansion when and only when they are
equivalent to a fraction in which the prime factorization of its
denominator consists only of 2s and 5s. In the latter situation, the
terms in the reduced form of such fractions can be raised so that the
denominator is a power of ten: ten, one hundred, one thousand and so
on. Whence the fraction has a finite decimal representation.
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Infinite repeating decimals. Many fractions after removing
common divisors in their numerator and denominators have denominators
with prime factorizations that include primes other than 3 and 5.
Those fractions do not have finite decimal expansions. But long
division leads to infinite decimal expansions that repeat - are
periodic. The expansion actually represent a sequence of finite
decimals, each with one more decimal place than the one before it.
The infinite decimal expansion of a fraction to 1, 2, 3, 4 and more
decimals places gives a set of decimal approximation to the fraction
- the latter could be improper. For example, the improper fraction \[
\frac 5 3 = 1 \frac 23 \] has decimal expansion 1.66666 with the 6
repeating. Here 1.6 is within 1 tenth of $\frac 53$ while 1.66 and
1.666 and 1.6666 are, respectively, within one hundredth, one
thousandth and one-ten thousandth of a unit from the fraction $\frac
53$ to the fraction as the number of decimal places taked in the
approximation increases. Repeating decimal expansions may also be
called periodic decimal expansions. With the help of a geometric
sum formula, or with the help of arithmetic with infinite decimal
expansion, one may calculate associated fraction.
-
Non-repeating decimal expansions correspond to points on the
line segment or half-line that are not whole number times a half, a
third, a quarter nor any other numerator-1 fraction multiple of the
unit length. For some it is an easy matter to imagine an infinite
non-repeating decimal expansion like
5.67920120492... [not all digits shown]
locates a point on the half-line as the limiting position of
decimal fractions, each fraction have one decimal place that its
predecessor. Here each decimal fraction determines a point on the
half-line. Non-repeating decimal expansions may also be called
non-periodic decimal expansions.
The Pythagorean theorem and consistency -assumed- of number theory
implies the length given by square root of 2 multiple of a unit
length exists and can not be given exactly by a fraction. Geometry
and the Pythagorean theorem shows given a whole number which is not a
square of another whole number, those whole number include all primes
3,5, 7, 11, 13 ..., that we can draw a right triangle with such the
square on one side has area equal to the whole number. But the side
of that square cannot be a improper nor a proper fraction times a
unit length.
Yet the notion of infinite, non-repeating decimals and arithmetic in
the limit with them allows geometric lengths and positions along a
half line to be given by finite and infinite decimal expansion,
repeating or not. In practice we use symbols like $\sqrt 2$ and $\pi$
with the understanding (i) they have or are given by infinite decimal
expansions; and (ii) figuring will be approximate and only employ
finitely many decimal places, the more the better for what we hope is
greater accuracy. The reason for that requires further knowledge of
mathematics. Some subtleties remain to be discussed.
The need for infinite decimals that are non-repeating may be surprising.
Counts are done with whole number - no fractional part. Measurements are
typically done to the nearest whole unit or nearest tenth, hundredth and
even thousandth of a unit in metric system, and to the nearest half,
third, quarter or other numerator-1 fraction of a unit - tenths and
hundredths, included. In counting exactly and in measuring to the nearest
numerator-1 fraction of a unit infinite decimal expansion are avoided.
Thus the common use of mathematics does not require infinite,
non-repeating decimals. With that, figuring is done with exact count,
exact measures and approximate measures.
Most people, scientists and engineers may work with measurements and
calculations that are approximate, albeit greater accuracy is preferred
for the sake of error control. Usually, small errors are acceptable
provided they can be controlled - made neglible - by doing calculation to
enought decimal places. Continuity of calculations helps with that. Thus
when we see symbols like $\sqrt 2$ and $\pi$ in formula, remember that in
practice, evaluation will be based on approximation of these and further
numbers. When we employ a calculator to evaluate a formula with these and
further symbols for numbers, we are doing an approximate calculation.
Experience may say or suggest how many decimal place need to be kept or
carried through a calculation to control or minimize error. Beyond that,
talk of real numbers that are irrational, that is, are not given by whole
number nor fractions is a theorectical construct. In essence, unsigned
real number are provided by whole numbers, by improper fractions as is or
in mixed number form, and by infinite non-repeating decimal expansion. A
few of the latter are given by square and n-th roots of whole numbers and
fractions where the latter are not squares or not fractions to the n-th
power.
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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.
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