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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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Ratio of Decimal Fractions
A ratio of decimal fractions is a compound fraction
in which the numerator and denominator are both decimals -
proper or improper fractions expressible as whole number over a power of 10.
Imagine for some reason, the question how many times 0.23
meters went into the distance 4781.55 meters was met. The division algorithm
above applies only to whole number divisors. So to obtain whole numbers, we ask how many
times does 23 centimeters = 23 x (0.01 meters) go into 4781.55 meters = 478155
centimeters. The long division of 478155 by 23 gives 20789 times completely with a
remainder of 8 centimeters = 0.08 meters leftover.
Shifting the Decimal Point
In general if M and N are given by decimal fractions then
for some whole or natural numbers P, Q, a and b. Here we assume
a and b give the number of digits after the decimal point in the finite decimal
notation for M and N respectively, and we may also assume the ones digit in both
P and Q are nonzero. Now
M
N |
= |
P
10a
---
Q
10b |
= |
P
10a |
* |
10b
Q |
= |
P*10b
10a |
* |
1
Q |
= |
P* 10b
10a
-----
Q
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|
The foregoing justifies the long division method of shifting the
decimal point in the dividend and divisor by number of decimal places in
the divisor to obtain an integral (whole number) divisor Q
Another Twist:
M
N |
= |
P
10a
---
Q
10b |
= |
P
10a |
* |
10b
Q |
= |
P*10b
Q*10a |
The twist yields
456.89 456.89 x 1000 456890
------ = -------------- = --------
34.567 34.567 x 100 34567
Long Division - Remainder Analysis and Convergence
The equalities
M
N |
= |
P
10a
---
Q
10b |
= |
P
10a |
* |
10b
Q |
= |
P*10b
10a |
* |
1
Q |
= |
P* 10b
10a
-----
Q
|
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imply for the long division computation of M/N
that we can shift the decimal point in both the denominator or
divisor N and dividend or numerator M to obtain an equivalent fraction in which
the denominator is a whole number Q.
Now for any whole number k, long division
10k P = s(k) Q+ r
where 0 < r < Q is a natural number.
Therefore division by 10k Q gives
P
Q |
= |
s(k)
10k |
+ |
1
10k |
* |
r
Q |
Here
Therefore
provides at least the first k digits of the decimal expansion of
P
Q
beyond the decimal point. The result
P
Q |
= |
s(k)
10k |
+ |
1
10k |
* |
r
Q |
can also (I presume) be obtained by continuing the long division process as
well.
Remark: The foregoing implies the decimal expansion of P/Q will either terminate
or provide an increasing Cauchy sequence of decimal fractions s(k)/10k
which converge to a limit.
Exercise: Show computing P/Q to k decimal after the decimal point gives
to k+b-a places after the decimal point. | |
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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