6. Number One and Decimal 0.9999 - 9 repeating
Each whole number and each decimal fraction has two decimal
representation, one with finitely many decimal places, and one obtained
by by subtracting one from the non-zero digit with smallest place, and
then adding a sequence of nines in the following decimal places. That is
the general idea - not precisely said. But a few examples may illustrate
the concept
The decimal fraction 34.806 has infinite decimal
expansion 34.80599999
(9 repeating)
The decimal fraction 874.4 has infinite decimal
expansion 874.399999 (9
repeating)
The decimal fraction 1.628 has infinite decimal
expansion 1.62799999 (9
repeating)
The whole number 1 has infinite decimal
expansion 0.99999 (9 repeating)
The whole number 4380 has infinite decimal
expansion 4379.99999 (9
repeating)
The whole number 4568098000 has infinite
decimal expansion
4568098999.99999 (9 repeating)
Special case of the number 1
The number 1 can be represented exactly by itself. It can also be
regarded as the limit of the sequence
0.9 0.99 0.99 0.999 0.9999
where the q-th term of the series is given by
and equals the finite decimal 0.999 ... 9 with q nines after
the decimal point.
The sequence
0.9 0.99 0.99 0.999 0.9999
is denoted, represented or implied by
The foregoing non-terminating decimal expansion which represents a
sequence of proper decimal fraction approximations to 1, that has
the value 1 as it limits.
Calculus Students: See Chapter 14 in Volume 3,
Why Slopes and More Math, for a or the decimal viewpoint of limits as a
form of decimal approximation in which error control is important in
either practice or principle, and possibly both.
Conclusion
The number one has two decimal expansion, the single digit
1 which is an exact representation, and the sequence
|
1 -
|
1
-----
10q
|
= 0.999 ... 9 with q nines
|
|
|
after the decimal point.
|
which converges to it in the sense that pth term (p>q) is guaranteed
to be with 10-q units of 1. The
sequence is denoted by
A Further Conclusion >
All decimal fractions, numbers of the form
where M is a whole number can be approximated by one and only
one infinite decimal expansion or or sequence of period one in the
the digit 9. Exercise prove this. The following
observations may help.
The number
is the limit of that sequence. Furthermore, the whole number M has
a unique finite decimal representation - two The decimal expansion
of
M
-----
10k
obtained by adjoing a decimal point to the decimal representation
of M and then shifting that decimal point k places to the left,
is also unique.
Remark: if a decimal expansion ends in recurring nines, we can replace it
by its limiting value - a finite decimal expansion - and use the limiting
value in our further calculations.
Problem: If two terminating or non-terminating decimals
differ at the k-th place in their expansion, and a least one does not end
in 9 recurring, then the two decimals expnasions represent
different numbers. Explain or show why.
Enriched Topic - Repeated
A Cauchy sequence f(n) has the following decimal
property: For each whole number k, there is a whole number
N with the following property: all terms in the sequence after
the first N-1 agree with each other
to at least k decimal places. This property allows us to define
and compute in principle an infinite decimal expansion. This expansion
is assumed to define a unique real number: the limit L of the
Cauchy sequence.
The limit of a repeating non-terminating, infinite periodic decimal
expansion is given by a simple fraction. For a proof, study the
geometric series or study arithmetic with infinite and/or periodic
decimal expansion
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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
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The Logic of Injustice:
How Texas sent
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
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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
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Mathematics
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Skills with take
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Arithmetic
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Algebra
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Geometry
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More
Algebra
70
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
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bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
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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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