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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
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
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
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.
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.
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Long Division Continued
calculation of places after the decimal point
Long Division method provides a sequence of decimal approximations to a
fraction M/N. If the sequence is finite the fraction is decimal. So if the
sequence does not terminate, the fraction is non-decimal.
20789 | 20789
---------- | --------
23 | 478155 | 23 | 478155
- 455000 as 23 x 2 = 46 --> 20000 | 46 (23 x 2 -> 2)
------ | - --
18155 | 18
-16100 23 x 7 = 161 --> 700 | 00 (23 x 0 -> 0)
----- | ---
2055 | 181
-1840 23 x 8 = 184 --> 80 | 161 (23 x 7 -> 7)
----- | ---
215 23 x 9 = 207 --> 9 | 205
-207 | 184 (23 x 8 -> 8)
---- | ---
8 Last Leftover or | 215
remainder is 8. | 207 (23 x 9 -> 9)
| ---
| 8 (less than 23)
| Stop.
|_____________________________
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The rightmost column shows the usual long division
algorithm, one that was taught in elementary school in the 1960s.
Variations of it may be found. You should be able to see how the steps on
the right correspond to those on the left. In the rightmost column, you
will see rows with * in them. The row in-between in them is usually
omitted to lessen the amount of writing. The row is included here to help
in the comparison of the Euclidean Division Method and that which I met in
elementary school in the 1960s. The right hand column is a more cryptic
implementation and variation of the Euclidean Division Method.
8
Conclusion 478155 = 20789 x 23 + 8 = (20789 + -- ) x 23.
23
Continuing the Division Process -
The remainder 8 is smaller than 23, but 3 x 23 = 69. So
80 = 23 x 3 + 11 and therefore, dividing by 10, yields
8 = 23 x 0.3 + 1.1
--------------------
This gives
478155 = 20789 x 23 + 8
= 20789 x 23 + 23 x 0.3 + 1.1
= 20789.3 x 23 + 1.1
The remainder has become 1.1 instead of 8. It is much smaller.
We can do this again, and again. For example: 4 x 23 = 92. So
110 = 23 x 4 + 18
and therefore division by 100 gives
1.1 = 23 x .4 + .18
This yields again
478155 = 20789 x 23 + 8
= 20789.3 x 23 + 1.1
= 20789.34 x 23 + .18
The remainder has become smaller. This division process can be
recorded in the shorthand form as follows
_
20789.34
---------
23 | 478155.0000
- 455000 as 23 x 2 = 46 --> 20000
------
18155
-16100 23 x 7 = 161 --> 700
-----
2055
-1840 23 x 8 = 184 --> 80
-----
215 23 x 9 = 207 --> 9
-207
----
8.0
-6.9 23 x .3 = 6.9 --> .3
---
1.10
-.92 23 x .0492 = .92 --> 4
-----
.18
478155 = 23 x 20789.34 + .18
where 0.18 = the remainder. The remainder approaches zero
as more and more decimal digits after the decimal point are computed
via the long division method.
Cosmetic Remark: In the division method
above, the decimal point in 20789.3408 is needed while those in the
computations below are usually omitted.
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www.whyslopes.com
Number Theory
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
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Long Division Continued
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
How to Add Reals
How to Multiply Reals
Distributive Law for Reals
Remainder Arithmetic II
Related Site Pages:
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