|
YOU are better than YOU think. Show yourself how:
|
-/[]\- 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;
|
-/[]\- 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. |
Ratio of Decimal FractionsA ratio of decimal fractions is a compound fraction
in which the numerator and denominator are both decimal fractions.
Shifting the Decimal Point In general if M and N are given by decimal fractions then
for some whole or natural numbers P, Q, a and b. Here we assume a and b give the number of digits after the decimal point in the finite decimal notation for M and N respectively, and we may also assume the ones digit in both P and Q are nonzero. Now
The foregoing justifies the long division method of shifting the decimal point in the dividend and divisor by number of decimal places in the divisor to obtain an integral (whole number) divisor Q Another Twist:
The twist yields 456.89 456.89 x 1000 456890 ------ = -------------- = -------- 34.567 34.567 x 100 34567 Long Division - Remainder Analysis and ConvergenceThe equalities
imply for the long division computation of M/N that we can shift the decimal point in both the denominator or divisor N and dividend or numerator M to obtain an equivalent fraction in which the denominator is a whole number Q. Now for any whole number k, long division
where 0 < r < Q is a natural number. Therefore division by 10k Q gives
Here
Therefore
provides at least the first k digits of the decimal expansion of P beyond the decimal point. The result
can also (I presume) be obtained by continuing the long division process as well. The foregoing implies the decimal expansion of P/Q will either terminate or provide an increasing Cauchy sequence of decimal fractions s(k)/10k which converge to a limit.
|
Related Site Pages:
|
|
|