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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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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.

Appendix E
Properties of Limits

Error Control Inequalities

Here suppose C is an approximation to c and D is an approximation to d. Further suppose |C-c| £ e₁ and |D-d| £ e₂. You could imagine that e₁ and e₂ are both numbers of the form [1/2]10^-k. Alternatively, you could imagine that C and D are estimates (from measurements perhaps) and that the true values of c and d lie in the intervals [C-e₁,C+e₁] and [D-e₂,D+e₂], respectively. Here the values of the errors would depend on circumstances beyond the control of a mathematics book. Now if the uncertain approximations C and D are used in computations in place of the true values c and d, there will be an error. Thus error control in computations will be of interest.

The following theorem describes the maximum possible error in the approximations C+D, C·D, [(C)/(D)] and [1/(D)] to the true values of c+d, c·d, [(c)/(d)] and [1/(d)].

Theorem E.1 [Error Control Inequalities] Assume |c-C| £ e₁ and |d-D| £ e₂. Suppose k is a real number. Then

|(C+D)- (c+d)| £ e1+e2 |CD-cd| £ |C|e2+|D|e1+e1e2 |kC-Kd| £ |k|e1

Moreover if D^* = |D|₂ > 0 then 

ê ê ê C D - c d ê ê ê £ |C| D* e1+ |C| |D|D* e2

and

ê ê ê 1 D - 1 d ê ê ê £ |C| |D|D* e2.

Only the values of the approximations C and D will be available unless c and/or d are known exactly. Because of this, the above maximum error estimates stated above involve the two estimates C and D of c and d, and not the exact values of the latter. But the derivation of these inequalities, show that the error estimates also hold with c and d in place of C and D. The conclusions would still hold if c and d were regarded as approximations to C and D.

The conclusion indicates the maximum possible absolute error in the calculation of sums, products, quotients and divisors. Roughly speaking, if the error was known precisely then the approximation could be improved to eliminate the error in them. Thus with the given information about the approximations, the above inequalities cannot be improved.

The above inequalities and a different perspective on them can be found along with proofs in the first chapter in the book Calculus by L. Bers (Holt, Rinehart and Winston 1969, SBN 03-065240-5). The chapter in question provides further background information on the decimal and decimal-free representation of real numbers.

In the direct use of the above inequalities or error estimates, the maximum possible errors e₁ and e₂ in the approximations are specified first. In the indirect use of the above inequalities, or error estimates, we may specify a maximum allowable (target) error e > 0 for a computation first, and then try to choose e₁ and e₂ second, so that the maximum allowable error is not exceeded. For example in the estimation C+D of the sum c+d, the error bounds e₁ ³ 0 and e₂ ³ 0 should satisfy

e ³ e1+e2

so that the actual error |(c+d)-(C+D)| £ e. The proof and the next pages may further clarify this error control effort.

Proof of the Inequalities.

For the sum, observe

|(C+D)- (c+d)| = |(C-c)+(D-d)| £ |(C-c)|+|(D-d)| £ e1+e2

For the product, observe

|cd- CD| = |(C+(c-C))(D+(d-D))-CD| = |[CD+(c-C)D+(d-D)D+(c-C)(d-D)]-CD|    = |(c-C)D+(d-D)D+(c-C)(d-D)]| £ |(c-C)D|+|(d-D)D|+|(c-C)(d-D)| £ |c-C|·|D|+|d-D|·|D|+|c-C|·||d-D| £ e1·|D|+e2·|D|+e1e2

For the reciprocal, observe |d-D| £ e₂ implies

|d| > |D|-e2 = D* > 0

and therefore [1/(|d|)] > [1/(|D|-e₂)] = [1/(D^*)]. Therefore

ê ê ê 1 d - 1 D ê ê ê = ê ê ê (d-D) Dd ê ê ê £ e2 |D|·|d| £ e2 |D|D*

For the quotient observe [1/(|d|)] > [1/(|D|-e₂)] = [1/(D^*)] implies

ê ê ê c d - C D ê ê ê = ê ê ê cD-Cd dD ê ê ê = ê ê ê [C+(c-C)]D-C[D+(d-D)] dD ê ê ê = |(c-C)D-C(d-D)| |Dd| £ |(c-C)D|+|C(d-D)| |Dd| £ |c-C|·|D|+|C|·|d-D| |Dd| £ e1|D|+|C|e2 |Dd| £ e1|D|+|C|e2 |D|D*

 

www.whyslopes.com
Real Analysis - Decimal View

Here are the Appendices from  Volume 3, Why Slopes and More Math,  Chapters 14 to 19 in Vol 3 are related. Here is a  reference for college or university mathematics, electrical engineering and physics.

A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

The site area More Calculus contains a one-sided theorem with proof that should be of interest too.

Vol 1A Logic Postscripts
online only:-

Proof by Absurdity alias proof by contradiction

How the demand for consistency supports the law of the excluded middle

Reality versus or with the aid of Imagination

Science, Engineering & Math Students: Have you seen a simpler  geometric introduction to complex numbers? ( java applet included) . Can you explain what is a variable without using a symbol? Can you derive trig expression for dot & cross &  cosine law from complex number properties? For truth tables and indirect methods of reason, see  chapters 19-24 & postscripts in  Pattern Based Reason  and visit Volume 1A, Pattern Based Reason, striving for objectivity, the empirical challenge & limits.  

Vol 1A Postscripts
online only

Proof by Absurdity alias proof by contradiction

How the demand for consistency supports the law of the excluded middle

Help Me Learn/Teach;

1. Algebra
   words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)
2. Arithmetic
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   - videos on primes, lcm, gcm,lcd, square roots etc
3. Calculus - geometric preview, algebraic preview,
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4. Complex numbers
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5. Education
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6. Fractions
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7. Functions - introduction
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8. Geometry, Euclidean - Correspondence of triangles,  Triangle construction,  duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
   
9. Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
10. Logic
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     Occurrence & Truth Tables - Indirect Reason -Indirect Reason More
11. Proportionality
    - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions
12. Real Analysis
    - Decimal View of concepts and of proofs

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