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Employ an online or offline tutor at your own risk from
AU:
tutorfinder.com.au
CDN :
findatutor.ca
CDN: .i-tutor.ca
CDN: Montreal
Tutors
NZ: findatutor.co.nz
UK:
tutorhunt.com
UK: tutors4me.co.uk
USA: wiziq.com
USA: ziizoo.com
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YOU are better than YOU think. Show
yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful,
Edifying, Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens
eyes. Leads to greater precision.
in reading and writing
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice is
approximately correct, for some circumstances, not all. Site How-TOs
are logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site
area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
For online automated help in senior
high school maths & calculus, visit quickmath.com
For Automatic Calculus and Algebra Help with derivatives, integrals,
graphs, linear equations, matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different
range of services, some free, some not, all based on webmathematica.
Good luck.
|
Explore collaborative whiteboards from groupboard,
twiddla or
scriblink.
|
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Decimal Expansions of the Number 1.
Related pages: Fractions
Fractions as Decimals
1 = 0.999 Recurring
Unsigned Reals Numbers
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
_
0.999
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.
So 1 has two decimal expansion, itself exactly and the sequence
| 1 - |
1
-----
10q |
= 0.999 ... 9 with q nines |
|
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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
_
0.999
Remark: All decimal fractions, numbers of the form
where M is a whole number can be approximated by one and only one
non-terminating 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.
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www.whyslopes.com
Number Theory
[ Back ] [ Up ] [ Next ]
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Place Value Reinforcement
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II
Related Site Pages:
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck
Food for thought: Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice..
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