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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. |
Decimal Expansions of the Number 1.
The number 1 can be represented exactly by itself. It can also be regarded as the limit of the sequence 0.9 0.99 0.99 0.999 0.9999 where the q-th term of the series is given by
and equals the finite decimal 0.999 ... 9 with q nines after the decimal point. The sequence 0.9 0.99 0.99 0.999 0.9999 is denoted, represented or implied by
The foregoing non-terminating decimal expansion which represents a sequence of proper decimal fraction approximations to 1, that has the value 1 as it limits.
So 1 has two decimal expansion, itself exactly and the sequence
which converges to it in the sense that pth term (p>q) is guaranteed to be with 10-q units of 1. The sequence is denoted by
Remark: All decimal fractions, numbers of the form
where M is a whole number can be approximated by one and only one non-terminating decimal expansion or or sequence of period one in the the digit 9. Exercise prove this. The following observations may help.
Remark: if a decimal expansion ends in recurring nines, we can replace it by its limiting value - a finite decimal expansion - and use the limiting value in our further calculations. Problem: If two terminating or non-terminating decimals differ at the k-th place in their expansion, and a least one does not end in 9 recurring, then the two decimals expnasions represent different numbers. Explain or show why. |
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