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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 FractionsFinite or Infinite & PeriodicThe long division process applied to a simple fraction
will terminate before or after the decimal point and give a whole number or a decimal fraction in decimal form if the simple fraction is equivalent to a decimal fraction. The long division will not terminate if the simple fraction
is not equivalent to a decimal fraction. In this case, long division in the first gives
where the remainder r is a whole number between 1 and N-1. The remainder r will be nonzero as we are not in the decimal fraction case. Continuing the long division process to q decimal places after the decimal point gives
where the q-th remainder rq is a whole number between 0 and N and is kq is a whole number..
Pigeon Hole Principle: Now the leading N remainders rq where 1 < q < N sequence of remainders takes values in the interval [1, N-1] of whole numbers - N-1 possible values in all. Therefore the remainders must repeat. If not, the interval [1, N-1] would include N distinct values. It follows that the decimal expansion of M/N must start repeating on or before the N first place. So the period of the decimal expansion is N or less.
Now for each whole number q of decimal places, the equation 10q *M = kq*N + rq implies
where rq is in the interval [1,N-1] repeats and eventually has period P in q, that is, rq+P = rq for q sufficiently large, and where km for m > q agrees with kq to q decimal places. Therefore, M has a periodic, non-terminating decimal expansion aTaT-1 ....a1.b1b2b3b4b5b6 .... in which the first q decimal places are provided by the decimal fraction
We assume that a non-terminating decimal expansion may be associated with a single point on a coordinate (half) line: Exercise: If two non-terminating periodic decimals differ at the k-th place in their expansion then the two decimals converge to different limits. Explain or see why.
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