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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 Fractions
Finite or Infinite & Periodic
The long division process applied to a simple fraction
M
N
will terminate before or after the decimal point and give a whole number or a
decimal fraction in decimal form if the simple fraction is equivalent to a
decimal fraction.
The long division will not terminate if the simple fraction
M
N
is not equivalent to a decimal fraction. In this case, long
division in the first gives
M = k N +r
where the remainder r is a whole number between 1 and N-1. The
remainder r will be nonzero as we are not in the decimal fraction case.
Continuing the long division process to q decimal places after the decimal point
gives
10q *M = kq*N + rq
where the q-th remainder rq is a whole number between 0 and N and
is kq is a whole number..
Now the infinite sequence of remainders takes values in the interval [1,
N-1] of whole numbers - N-1 possible values in all. Therefore the remainders
must repeat. If not, the interval [1, N-1] would have infinitely values.
Pigeon Hole Principle: Now the leading N remainders rq
where 1 < q < N sequence of remainders takes values in
the interval [1, N-1] of whole numbers - N-1 possible values in all. Therefore
the remainders must repeat. If not, the interval [1, N-1] would include N
distinct values.
It follows that the decimal expansion of M/N must start repeating on or
before the N first place. So the period of the decimal expansion is N or less.
Check Logic later: Am I misleading myself in the above argument?
Now for each whole number q of decimal places, the equation
10q *M = kq*N + rq
implies
M
---
N |
= |
kq
-----
10q |
+ |
rq
-----
10q |
where rq is in the interval [1,N-1] repeats and eventually has
period P in q, that is, rq+P = rq for q
sufficiently large, and where km for m > q agrees with kq
to q decimal places. Therefore,
M
N
has a periodic, non-terminating decimal expansion
aTaT-1 ....a1.b1b2b3b4b5b6
....
in which the first q decimal places are provided by the decimal
fraction
We assume that a non-terminating decimal expansion may be associated with a
single point on a coordinate (half) line:
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 decimal expansion is
given by a simple fraction. For a proof, study the geometric series.
Exercise: If two non-terminating periodic decimals differ at the k-th
place in their expansion then the two decimals converge to different limits.
Explain or see 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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