1. The Counting Origins of Numbers
Students need to know how to count. A knowledge of how
counting might of began with tally marks on tally sticks (or walls) may
provide a context.
Tally marks on a stick may have given the earliest way for an individual
to keep track of how many objects possessed or owed.
As long as there is a one-to-one pairing, matching or correspondence
between the tally marks and the objects being tracked, the owner assumes
none have been lost or gained. The tally marks altogether describe and
visually count or keep track of how many objects there are.
For example, a shepherd may put a tally mark on the tally stick above
for each sheep that enters a pen, and later on verify as the sheep
leave the pen that there is still one mark per sheep. That keeps track
of the sheep and ensures none have been lost nor gain. Births and
predators are assumed to account for all changes in the correspondence.
If the tally marks are all alike as they are made, the order in which
sheep leave the pen may be different from then one the entered. All
that is important in concluding that no sheep has been gains or lost is
that each sheep on exit be in one to one correspondence with a tally
mark and that all tally marks are used.
In pure mathematics, two sets are said to be have the
same cardinality (count) or to be equipollent when and only
when there is a bijection or one-toone-correspondence between
them. A bijection in brief is a one to one pairing between the
elements of one set and the elements of the other, so all elements in
each set belong along to one and only one ordered pair.
Tracking sheep or marbles with tally marks on a stick (or paper)
provides a bijection between the set of tally marks and the set of
sheep.
The shepherd in keep track of his sheep may have tally marks on several
sticks. In that case, the sheep may be allowed to exit the pen via
several exits. Again as long this exiting gives a one to one pairing
between the sheep and tally marks, the shepherd and we assume none have
been lost. The sheep could be tallied in the first instant by
entering the pen via several entrances or in several groups. It
is possible to allow the sheep to enter grouped in one way and to leave
grouped in another way. And if we tallied the entering and
leaving each day using different tally ticks, and different groupings
of the sheep and tally sticks, the different tallies and the
sheep should be in one to one correspondence.
Working Definition of Whole Numbers: A set of tally marks on
a stick gives a whole numbers. So whenever we speak of a whole number, we
think of a set of distinct tally marks on a real or imagined stick.
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Tallying or Enumeration Assumption: If a set is tallied
(counted) in two different ways, with or without the use of
grouping, the set of tally marks for one way will lie in a
bijection (one to one correspondence) with the marks in the other
way.
In other words, we assume any two ways to count the elements of a
set will result in the same number. Later on, we will
see that the equality of two ways to count or measure what is a set
or region leads to properties of arithmetic.
First Tallying or Enumeration Assumption: For a
person who can count, the number of tally marks will be the
same regardless of the order in which the elements of a set are
tallied or counted.
Collecting the tally marks into groups provided a counting or
numeration. Today, we go further. In decimal notation we use
digits 1 to 9 to as marks for single object or a group of them, and
we use place value to keep track of how ones, tens, hundreds,
thousands etc there are in a count. So we track groups of
power of ten instead of individual elements of a set or sheep in a
pen. None the less, tallying goes on. And when we are counting
elements of a set, we assume the final count or the decimal for it
will be independent of how and in which order the set elements are
tallied, counted or numbered.
Decimals
Decimal notation provides a compact form of tally marks for
tallying or counting or describing how many objects there are in a
set. The decimal view of numbers and number theory may be read
parallel to the general number theory pages.
The Start of Number Theory Continues with the
following pages
Adding Wholes Multipling Wholes Distributive
Law Preamble
Distributive Law for Wholes Consequences More Consequences
What is a Fraction Compound Fractions
after this one (and not the next links in top and bottom
margins
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Remark for Students of Pure
Mathematics: The following gives a set
theoretic justification of the unique tallying or
enumeration principle
Two sets are equipotent when and only
when there is a one-to-one map from one onto the
the other. The latter notion works for
both finite and infinite sets as
well.
Theorem: If there is a one-to-one
map f of a set U into a proper subset W of itself then
the set is not finite:
Proof If U has N elements and
we can find a one-to-one labeling of those N elements u
of U given by a map c(j): [1,N] -> U.
Now d(j) = f(c(j)): [1,N] -> W, a proper subset of
U. Therefore, there is an element s in U not W. Pick
such an s and Put d(N+1) = s. Then the
extended map d: [1,N+1] -> U has a range
with N+1 elements in U. The foregoing shows that if we
count N distinct elements in U with the aid of the
mapping c then with the aid of the mapping
f, we can also count N+1 elements as well using another
function d. Therefore for all whole numbers
N, there is no bijective map of a range of whole
numbers 1 to N onto U. That is what we mean by saying
the set U is finite.
Contrapostive form of theorem: If U is a
finite set (that is, is in bijective correspondence
with a set of whole numbers 1 to N) then then any
one-to-one map of U into itself must be
surjective. - if it was not then U would be
not finite, that is infinite.
Unique Tallying Theorem: If P is a finite
set with two maps c: [1,N] --> U and d:[1,M]
--> U are bijections of two possibly different
intervals [1,N] and [1,M] then N = M.
Proof: If N < M then for each j in the
interval [1,M], there is a unique u = d(j) in P
and hence a unique k = c-1(u) =
c-1( d(j) ) = f(j) in [1, N] and hence in
[1,M] Therefore f:[1,M] --> [1,M] is an
injection. Yet [1,M] is finite. So the injection must
be surjective and hence N = M. The
alternative possibility M < N is treated
similarly. Q.E.D.
Note each of the maps in the Unique Tallying theorem
represents a way to count the number of elements in the
set P
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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
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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
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Parent Center: Help your child or teen
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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
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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;
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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
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Geometry
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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