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YOU are better than YOU think. Show
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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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Chapter 18
Rules for Algebra
Previous Real Numbers and
Quantities
Arithmetic Rules and Patterns
What They Do. The rules of arithmetic say when the order of operations
can be changed in a first calculation, so that we obtain a second calculation
which gives the same result as the first. These rules apply to arithmetic
involving real numbers and/or real quantities.3
3
High school mathematics (circa 1990) talks only about real numbers, and leaves
talk about quantities to physic courses and commerce courses. But calculations
involve both real numbers and units of measurements. The convention in algebra
textbooks is to emphasize the connection with real numbers but not real
quantities. But in dealing with quantities in physical and monetary
calculations, students need some guidance. Since the rules of algebra apply to
calculations involving units, an algebraic tradition involving the
manipulation of units and their powers needs to be presented and sanctioned in
high school mathematics courses.
The order of arithmetic operations, suggested by parentheses, matters in some
calculations, but there is some flexibility. In some but not all, we can change
the order in which arithmetic is done without changing the arithmetic result.
The properties of arithmetic (rules) given below say how this can be done.
Explaining Some Rules.
Next, you may meet more words than you ever wanted on the rules of
arithmetic. Read on and look for the ideas new to you. Some, just a few, could
be worth repeating to others.
To describe the properties rules for changing calculations without changing
their results, we introduce four shorthand letters a, b, c
and d to stand-in for real numbers (or real quantities). The use of these
letters is a tradition. Other letters could be used. Sometimes it is convenient
to describe or rewrite these rules or properties using other letters.4
You could pick four different letters if you wish.
4You
should imagine these rules written with other letters of your choice, when in
the calculations you meet, at least one letter a, b, c
and d, that has been previously assigned a different role or meaning.
In any plot, each actor should have only one role.
The following table describes properties of addition and multiplication which
you can use in doing arithmetic or describing arithmetic that could be done. In
these laws and properties, the expressions on either side of the equal sign,
always give the same result.
| Properties of Addition and
Multiplication |
| First expression |
= |
Second expression |
name of the property (or rule) |
|
(a+b)+c = a+(b+c)
|
associative law for addition |
|
(ab)c = a(bc)
|
associative law for multiplication |
|
(a+b)c = ac+bc
|
(right) distributive law |
|
c(a+b) = ca+cb
|
(left) distributive law |
|
a+b = b+a
|
commutative law of addition |
|
ab = ba
|
commutative law for multiplication |
|
a+0 = a
|
additive identity: the effect of adding zero |
|
a·1 = a
|
multiplicative identity: the effect of multiplying by one. |
In each row of the above table, the first expression always gives the same
result as the second expression, no matter what real numbers or quantities the
letters a, b and c represent. In describing a calculation,
either expression can be replaced by the other, or a symbol (pronoun)
representing the result of either calculation.
The above rules only involve addition and multiplication. We will talk next
about the above properties and rules and about how they are used, next. How to
apply these rules to expressions involving subtraction or division will also be
described later.
Reminder. The product a×b is also written as a·b
or as ab. Which notation is used to signal multiplication is a matter of
taste and convenience. When the times symbol × might be confused with the
letter x, remember to use the dot · instead, write a·b or
ab.
Remark. The above properties are assumed and used in doing arithmetic
and in changing and manipulating formulas. They are often called the laws of
algebra. This author prefers to call them laws or properties for arithmetic.
Chapter Sections: [ Up ] [ 18 Changing Formulas ] [ 18. Proper Use of Equal Sign ] [ 18. Replacement & Substitution ] [ 18 Real Numbers & Quantities ] [ 18 Rules for Algebra ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Sum Associative Property ] [ 18 Sums and Number 0 ] [ 18 Replacing Subtraction by Addition ] [ 18 Times Properties ] [ 18 Sum Grouping and Ordering ] [ 18 Product Associative Property ] [ 18 Products with the Number 1 ] [ 18 Product Grouping and Ordering ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ PS: Rules for Fractions and Division ] [ 18 Inconsistent Nttn ]
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[ Back ] [ Up ] [ Next ]
Three Skills for Algebra
www.whyslopes.com
Foreword, Chapters
& Appendices
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
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