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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. |
Chapter 4. Longer Chains of Reason
The principle of mathematical induction stated below describes the above (previous) ladder idea in the algebraic shorthand notation favored in mathematics. The last part of this chapter will not make sense to you if you are not familiar with this shorthand notation. If this is the case, you may skip this description of mathematical induction. If you read it, and you find that you do not understand it, you could return to it later after you have seen the following chapters on algebra. They explain the use of shorthand in algebra. Mathematical InductionWe assume that when or if we have counted to any number n, we can count to the next one as well. Just add one to the count n. This gives the next number in our count which is written n+1. This offers a way to begin counting all the whole numbers 1, 2, 3, 4 and so on.Suppose or imagine for each whole number n, there is a situation An. This gives a step on the ladder. Now the next whole number after a whole number n is given by adding 1, that is n+1. So the next step after An is written as An+1. The principle of mathematical induction says the following:
The word occurs can be replaced by the expression can be reached. The principle of mathematical induction is quite simple. It requires the following: (1) there is a ladder; (2) on the ladder, from each step we can reach the next; and (3) the first step is reachable. When these three requirements are met, the principle of mathematical induction says: all the steps can be climbed or reached. That is all there is to this inductive principle. Question. What can be said about the reachability of An where n ³ 4 if we find a ladder for which requirements (1) and (2) are met, and we somehow know (3¢) that A4 is reachable? Hint: Imagine a ladder where the first three steps are broken, but the fourth is somehow climbable. Is the ladder climbable?. Next Chapter: 5. Islands and Division of Knowledge |
Foreword, Chapters and Appendices follow.
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www.whyslopes.com
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