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Odds & Ends
Group I
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24. The 20 Times Table
25. Algebra Formulas
26. On Learning Maths
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More
Group II
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
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More Algebra Hints
The simplification is shown with or
without the use of primes. Here computation may equal not
the decimal approximation but the algebraic or cosmetic
simplification of square roots. The examples below show how
factorization and prime decomposition, together or not, may
be used in the simplification process and also providing a
stopping rule.
Real Player Videos give more examples. View
them before, after besides the text below.
Square Roots of Whole Numbers without a calculator
If you have a calculator, you may compute or represent the square root of a
number exactly or approximately. But in algebraic calculations (or shorthand
mathematical reasoning with letters and symbols), approximations are to be
avoided. The latter may be done using the following methods. Some of these
methods are cosmetic. But their use leads to a common or standard form for
expressions involving square roots.
Irreducible Case - Leave as is
If h is prime or a product of primes to the first power then no simplification of the square root
__
Ö h
is possible. Leave as is.
Reducible Case if h is a perfect square
If h = n2 and n > 0 then
Examples
__
Ö 9 |
= 3 |
|
___
Ö 25 |
= 5 |
|
____
Ö 169 |
= 13 |
|
Combined Case
h = a2b is a product of a perfect square a and another number b
which may or not be irreducible.
For a > 0 and b > 0,
__
Ö h |
= |
___
Ö a2b |
= |
a |
__
Öb |
Examples
____
Ö 500 |
= |
______
Ö (100)5 |
= 10 |
__
Ö 5 |
____
Ö 27 |
= |
______
Ö 323 |
= 3 |
__
Ö 3 |
_____
Ö 1200 |
= |
_______
Ö (100)12 |
= 10 |
___
Ö 12 |
___
But Ö 12 |
= |
____
Ö 223 |
= 2 |
__
Ö 3
|
Therefore
_____
Ö 1200 |
= 10 |
__
Ö12 |
= 10(2 |
__
Ö 3 ) |
= 20 |
__
Ö 3 |
Simplification Revisited
If h = a2b where the prime factorization of b only includes
primes, but no powers of primes (other than 1). Then
Example
h= 1500 = 500*3 = 3*22*53 = = 3*22*52*5=
(22*52) 3*5 = (2*5)23*5
gives
____
Ö1500 |
= |
2*5 |
___
Ö3*5 |
= |
10 |
__
Ö15 |
|
More Simplifications:
For a > 0, b > 0 and c > 0,
_____
Ö a2b2c |
= |
ab |
__
Ö c |
Example
_____
Ö 1200 |
= 10 |
_______
Ö100*4*3 |
= (10*2) |
__
Ö 3 ) |
= 20 |
__
Ö 3 |
Suppose h = a2b where the prime factorization of b only includes
primes, but no powers of primes (other than 1).
- [Play
Video] 5 minutes - Calculation of
Squares and Square Roots for Natural
Numbers without and with decimal
approximations. Exact representation of square
roots without approximation requires not using
a calculating. That is important in algebra -
the statement and derivation of formulas.
- [Play
Video] 1¾ minutes - How to Compute
Square Roots by Factorization
- [Play
Video] 3 minutes - Computational
Properties - More on square computation by
factorization.
- [Play
Video] 3 minutes - Examples of square
root computation by factorization.
- [Play
Video]3¾ minutes - Examples of
square root computation by prime
factorization.
In algebra, this simplification rewrites square roots in
a standard form, a standard that may lead to a common
representation of square roots of whole numbers when they
appear in formulas and the derivation or justification of
formulas.
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Professor Whyslopes:
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Two gaps
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appears with too few words of explanation in high school and college
mathematics. Online Volumes 2 and 3 offer remedies.
Chapters
8 to 12 in Volume 2 put more words into the explanation and
comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and
indirect use a formulas identifies a unifying theme for mathematics
and logic - all rules and patterns will be used forward and backwards.
Chapters
2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very
challenging use of algebra in the high level mathematics: calculus.
Calculus requires earlier high school mathematics at full strength: (i)
This logically complete but long lesson on complex
numbers shows how to simplify the senior high school
exposition of circular trig functions upto to formulas in the plane
for vectors dot and cross-products. The lesson provides the
route that would have been taken in course design if the key element
of the lesson, a December 2009 invention, had been available in
the 1950s. For further algebra skill development. See the site
coverage of fraction
with units, proportionality,
ratios and rates,
polynomials, quadratics
functions
and straight
line slopes and equations.
- The Arithmetic Gap: An exact and efficient
mastery of arithmetic with decimals and fractions is best (required)
for the high level study of mathematics alone and in science,
technology and business. Pages here on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill the gap. That
exact and efficient command should be obtained in the last years of
primary school and the first years of secondary school.
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Skill mastery in
mathematics has to be seen to believed. To that end,
learn or teach how-to write and draw the steps in mathematical
figuring or reasoning clearly. Do not try to save space
by doing a sequence of step in one place. Instead, do or record the
steps in sequence on a separate lines to make each step obvious and
verifiable.
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Online Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Math
How-TOs etc 2008
1. Arithmetic
2. Algebra
3. More
Algebra
4. Geometry
5. More
Geometry
6. Calculus
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Math Lessons
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Flash Videos 11-2008
2. Algebra Videos (to appear)
3. Fractions
and More
4.. Solving
Linear Equations 04-2005
5. Euclidean-Geometry
To Complex No.s
6. Analytic
Geometry/Functions 2006
7. Number
Theory. 2006-7
8. Exponents,
Radicals & logs. 2008
9 Calculus
2005
10..Real
Analysis 1995
11 Electric
Circuits Etc 2007
12. .Algebra,
Odds & Ends, HS level-2001
13.Maps,
Plans, Similarity &Trig, with
Complex Numbers, 12-2009.
For Math
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1. K0-11Applied Math Program Outline
2. Mathematics
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4. (150 pages)
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