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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 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. |
Inequalities: Reason for a Language ChangeThe mathematical usage of terms or words sometimes drifts away from the common usage. The gap or conflict here can be a source of misunderstanding. For ease of exposition, a change in the terminology or names employed may be required. Here is a change that may affect every textbook. Not having a difference between the meanings of greater than and more positive than is a source of confusion in the exposition of mathematics, an unnecessary barrier to comprehension. Occam's razor should be applied. Why Say "More Positive Than" Instead of "Greater Than"The concept of greater than or more than is understood by students when dealing with counts or whole numbers. Before the introduction of signs, that is negative and positive numbers, finite decimal expansions extend this idea of greater than or more than. A finite decimal expansion in particular counts the number of units, tenths, thousandths and so on that the number it represents can be divided into. Beyond this, students may be shown or pointed to the comparison of (unsigned) numbers with infinite decimal expansions, albeit such comparisons may be rare due to the prevalence in everyday computations of finite decimal representations and expansions. When dealing with unsigned numbers, the ideas of greater than and more than imply that the larger number can be obtained from the lessor number through the addition of an (unsigned) number. The two concepts agree for positive or unsigned numbers. With the introduction of positive and negative numbers and zero on say the real number line, the technical ideas of greater than differs from the common usage, or the introductory idea of comparison of by size or magnitude (apart from any signs that may be present). Because of this students are tempted to say that a real number a is greater than another real number b if the magnitude of a is greater than the magnitude of b. The latter means real number a is greater in magnitude than another real number b. The task is to remove the temptation or conflict. The symbol > traditional has been called the greater than sign. Technically, given two real numbers a and b we write a>b if and only if there is positive number c such that a = b + c. The tradition is read a > b as the statement that a is greater than b. To avoid confusion, and to align mathematical terminology with the common usage, the symbol > should be renamed the more positive than symbol. This new name corresponds precisely to the technical meaning. With this new convention, the phrase a greater than b can revert to the common usage and mean |a| > |b|. Similarly, a < b can be read not as a is less than b but as a is more negative than b. This new terminology means there is a positive number c such that a = b - c or equivalently such that a + c = b. The signs <= and >= now may be read as more negative or equal to and more positive or equal to. Linear and Nonlinear OrderingsA number b is said to between two other numbers a and c if and only if there is a positive number q < 1 such that b=qa + (1-q)c. Ordering of the real line by the relationship more positive than provides a linear ordering of the real line: for any three points a, b and c on the real line the relationships a < b <c imply that b is between a and c. Ordering by magnitude provides a linear ordering of the positive numbers. For any three unsigned (positive) numbers a, b and c, the relationship a < b < c implies that b is between a and c. But for any three points a, b and c on the real line the relationship |a| < |b| < |c| does not imply that b is between a and c. So ordering by magnitude (or absolute value) of points on the real line is nonlinear. And to return to the initial remarks, the statement a is greater than b should mean |a| = |b| + c for some positive number c
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