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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.
| |
Solutions for Problems 2.1 and 2.2
2.1 Basic Stuff
Perform the indicated calculations by hand. Then check your calculations with
the aid of a calculator.
- 456+76+312 = 844
- 176·86 = 15136
- 4892-2396 = 2496. Check: 2496+2396 = 4892
- 1416¸813 = 1.742 to 3 decimal places after the
decimal point.
- 2396-4892 = -(4892-2396)
= -(2496) = -2496
2.2 More Basic Stuff
Simplify if possible. Remember that operations inside parentheses ( ) or
brackets [ ] are to be done first.
| A = (4 ¸5)¸3
= |
é
ê
ë |
4
5 |
ù
ú
û |
× |
é
ê
ë |
1
3 |
ù
ú
û |
= |
4
15 |
|
|
| B = 4 ¸ |
|
5
3 |
= 4× |
é
ê
ë |
3
5 |
ù
ú
û |
= |
|
12
5 |
= 2.4 |
|
| C = 4 ×(5 ×3) = 4 ×(15) =
60 |
|
| D = (4 ×5) ×3 = (20) ×3 =
60 |
|
| E = (4 -
5) - 3 = (-1)-3
= -4 |
|
| F = 4 -
(5 - 3) = 4-(2)
= 2 |
|
| G = 4 -
5 -3 = 4-(5+3)
= 4-8 = -4 |
|
| H = |
|
__
Ö32
|
= 3 = the principal
square root. |
|
Here 32 = 3 ×3. The other square root of 9 is -3.
This question is perhaps ambiguous - oops, unless you we follow the convention
that the phrase square root" here means the principal square root. We will
follow this convention below.
| I = |
|
____
Ö(-3)2
|
__
= Ö9
|
= |
___
Ö(32)
|
= 3
|
|
in accordance with the convention that the square root of a positive number
is always taken to be its principal square root. The answer -3
is not acceptable according to this principal square root convention. The number
-3 is the other square root.
K =
|
|
______
Ö (42+32)
|
=
|
|
____
Ö16 +9
|
=
|
|
___
Ö (25)
|
= 5
|
|
L =
|
|
_________
Ö (42 + (-3)2)
|
=
|
|
__
Ö 25
|
= 5
|
|
Division by a fraction p/q gives the same result as multiplication by its
reciprocal q/p.
Therefore
|
|
|
| |
5
4 |
¸ |
é
ê
ë |
8
7 |
¸ |
9
5 |
ù
ú
û |
= |
5
4 |
¸ |
é
ê
ë |
8
7 |
× |
5
9 |
ù
ú
û |
|
|
|
|
|
| ( |
5
4 |
) ¸ |
é
ê
ë |
(8 ×5)
(7 ×9) |
ù
ú
û |
= ( |
5
4 |
) × |
é
ê
ë |
( 7 ×9)
(8 ×5) |
ù
ú
û |
|
|
|
|
|
| ( |
5
4 |
) × |
63
40 |
= |
(5 ×63)
(4 ×40) |
|
|
|
|
|
|
63
( 4 ×8) |
= |
63
32 |
= 1 + |
31
32 |
= 1.96875 |
|
|
exactly. Thus M > 1. Different ways of obtaining and writing the
answer are possible. All are permissible provided you knew the justification or
rule applied in each step of your figuring or reasoning steps.
The numbers appearing in the calculation of M are identical to those
appearing in the calculation of
|
|
|
|
é
ê
ë |
5
4 |
¸ |
8
7 |
ù
ú
û |
¸ |
é
ê
ë |
9
5 |
ù
ú
û |
= [ |
é
ê
ë |
5
4 |
× |
7
8 |
ù
ú
û |
× |
é
ê
ë |
5
9 |
ù
ú
û |
|
|
|
|
|
|
35
32 |
× |
5
9 |
= |
35 ×5
32×9 |
= |
175
288 |
= |
175
288 |
< 1 |
|
|
But the order of division is different. This change in or grouping of division
operations changes the result. Here N ¹ M.
The number
|
|
|
|
5
4 |
× |
é
ê
ë |
7
8 |
× |
9
5 |
ù
ú
û |
= |
5
4 |
× |
7 ×9
8 ×5 |
|
|
|
|
|
|
5 ×(7 ×9)
4 ×(8 ×5) |
= |
7 ×9
4 ×8 |
= |
63
32 |
|
|
|
In handwriting, the letter O looks too much like the number 0. To avoid
possible confusion with the number zero 0, the letter O should NOT be
used as a shorthand notation to represent a number.
The factors of the number O and the number
|
|
|
|
5 ×7
4 ×8 |
× |
9
5 |
= |
(5 ×7) ×9
(4 ×8)×5 |
|
|
|
|
|
|
5 ×(7 ×9)
(4 ×8)×5 |
= |
7 ×9
4 ×8 |
= |
63
32 |
|
|
|
are identical, but the ordering and grouping of multiplication is different. But
for multiplication of fractions the ordering and grouping of factors does not
affect the result of a computation.
is not defined. The meaning of the expression
is not clear. Should it represent the calculation
|
é
ê
ë |
5
4 |
¸ |
7
8 |
ù
ú
û |
¸ |
9
5 |
or |
5
4 |
¸ |
é
ê
ë |
7
8 |
¸ |
9
5 |
ù
ú
û |
? |
|
Each of these expressions has a different value.
R =
|
|
__
Ö16
|
__
+ Ö9 -
|
|
__
Ö25
|
= 4+3 - 5 = 2
|
|
| T = 3.1416 - |
22
7 |
= 3.1416 -
3.142857143 = 0.001257143 (approx.) |
|
Surprise perhaps, this answer T is nonzero as both 3.1416 and [22/7] are
different approximations to the same number p.
U = p- 3.1416 ¹
0 as p is not exactly
3.1416 A better approximation to p
is 3.141592654 but the latter is still not exact. The decimal expansion of p
is infinite and non-repeating as the number p is not
rational - why is a intellectual debt left to a higher mathematics course, if
any. Here not rational means p is not a number of the
form [(p)/(q)] where both p and q are whole numbers.
| V = |
|
_____
Ö42-52
|
= |
|
______
Ö( 16-25)
|
= |
|
___
Ö(-9)
|
|
|
This square root is not defined. The expression for V is another example
of our ability to describe calculations that might be done or not, might be
impossible to complete. The calculation of V becomes possible if you know
about Ö[(-1)] and the
complex numbers. | |
www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
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
|