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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes your attention. |
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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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Chapter 9
Talking about Numbers or Quantities
Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]
4 Approximate Knowledge
Numbers and quantities are known, given, measured or estimated with varying
precision. For instance, the cost of a hot dog could be 2.25 dollars. This cost
is given exactly. In contrast, the height of a man might be between 5[1/2] and 6
feet and the weight of a truck could be between one and ten tons. In these two
cases, the quantities in question are sandwiched or bracketed between two
extreme values: the least and greatest possible. (The term sandwiched is
preferred. It is more graphic.) The distance between the bracketing values
measures the uncertainty in our knowledge.
7NOTE
FOR ADVANCED STUDENTS: More precisely, if x is a number whose value is
known to be between two positive number a and b with a £
b, then the mean value c = [(a+b)/2] gives an
approximation to x. The absolute error in this approximation is £
[1/2]|b-a|.
The percentage error in this approximation is £
100·[1/2][(|b-a|)/(a)]%.
The relative error in this approximation is £
[1/2][(|b-a|)/(a)].
To say that the percentage error is at most 1% indicates a better
approximation than a percentage error of at most 5% or even 100%. In the above
examples, note for instance the following: The height of the man is known
within 100[(0.25ft)/(5.5ft)] = 4.55% £
5%, a small (?) uncertainty. The weight of the truck is known within 100[(4.5tons)/(1ton)]
= 450%. The uncertainty in the latter is large.The symbol £
is shorthand for the expression less than or equal to.
Knowledge of numbers and quantities may be exact or approximate. But we can
still speak about them. We can also use approximate values in calculations and
then hope the resulting error is not too large. Estimating errors in
calculations is a useful topic which cannot be fully explored here. Error
estimation is limited by the observation that perfect knowledge of the error in
a computation would provide a means for removing the error. So error estimates
must remain imperfect.
8Significant
Digits etc: When you say that the height of a building is 10.47 meters
(approximately) without giving any further information, the uncertainty in the
last digit 7 should be £ [1/2]. When a single
decimal is used to approximate a number or quantity, the digits in it are said
to be significant when and only when the uncertainty in the last digit written
is £ [1/2] of a unit. Digits which are uncertain
by more than [1/2] should not be written when we report the result of a
measurement or calculation.
Exception: When a single quantity x is
bracketed between two others, say a and b, their mean value c
= [(a+b)/2] provides an approximation to x with an
error of at most d = [(|b-a|)/2].
In this case we may write x = c±d
and keep some digits in the decimal expansion of c with an
uncertainty in them of more than one half unit. Writing x = (10.472±0.003)
meters for example provides more information about x than the single
estimate x = 10.47 meters.
In some situations, the location of the last digit with an
uncertainty of less than [1/2] of a unit may be unknown and this convention
may be difficult to follow. Errors in long calculations may be minimized if
rounding-off is postponed as long as possible, for instance done at the end of
all calculations and not for intermediate results.
Another Example: In crossing a toll bridge with one rate for
trucks weighing under 10 tons and with a higher rate for trucks over 10 tons,
the knowledge that the truck is between one and ten tons means that the lower
rate is used. But in crossing a bridge with a higher toll rate for trucks over
five tons, the knowledge that the truck is between one and ten tons is not
accurate enough. The truck has to be reweighed.
Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]
Next Section: 9- Numbers versus Quantities,
or what a difference a unit makes.
Next Topic: What
is a Variable:
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www.whyslopes.com
Vol. 2, Three Skills for Algebra
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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