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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.
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Fractions and Decimals
The site area Fractions, Ratios, Rates, Proportions & Units give a thought-based development of fractions from definition to performing arithmetic operations of addition, subtraction, divisions and multiplication efficiently. That development provides one logical base for the following and the assume properties of unsigned fractions. The site area Fractions, Ratios, Rates, Proportions & Units does not emphasize decimal fractions. The addition and multiplication of unsigned fractions is commutative and associative due to the corresponding properties of natural numbers established in the first pages of this site area. By converting a pair of fractions A and B to a common denominator, the distributive law (A+B)C = AC + BC follows from the distributive law for natural numbers established earlier in area pages. Simple & Decimal FractionsA simple fraction is given by a pair of whole or natural numbers
where N denotes the denominator and M denotes the numerator. Here N the denominator must be non-zero. The fraction may be proper or improper. In the case where N = 1, we identify the fraction with M. A decimal fraction has the form
where the numerator N = 10k is a power of 10 for some natural number k > 0. The decimal fraction is improper if M > N = 10k. A fraction which is not equivalent to a decimal fraction is called a non-decimal fraction. Identifying and Recognizing Decimal FractionsWhich Fractions are equivalent to decimal fraction Assume the numerator N of a simple fraction M is given by product of 2s and 5s, we have N= 2a5b for some natural numbers a > 0 and b > 0. So
We will consider three cases: (i) a > b and (ii) a = b and (iii) a < b and show in each case that the simple fraction is equivalent to a decimal fraction. In the first case (i) a > b, we have a series of equivalent fractions
So M/N is equivalent or equal to a decimal fraction. In the second case (ii) a = b, we have
and the original fraction is a decimal fraction. In the third case (iii) a < b, we have a series of equivalent fractions
So M/N is equivalent or equal to a decimal fraciton. Conversely if
is a decimal fraction then its denominator 10a = 2a5a is a product of 2s and 5s, and it may be possible to cancel some of the 2s and 5s if the latter occur in the prime decomposition of the numerator M.
Place Value Notation for Decimal FractionsWhen the numerator, the whole M in a decimal fraction
has a decimal representation, the decimal representation of the decimal fraction is given by moving the decimal point k places to the left. The latter is illustrated in the following examples when read them from left to right.
Counting Digits in the products of Decimal Fractions
If M and N are given by decimal fractions then
where the one digits in P and Q are nonzero then
has a+b digits after the decimal point. Moreover, if P and Q have r and s digits in total then M*N and P*Q have r+s digits in their decimal expansions. Further, M*N has (r+s) - (a+b) digits before the decimal point. The proof is left to an exercise. Assume the most significant digit in P and Q is nonzero
Location of decimal point in product in column method computationPreviously ,we met the calculation 823 The foregoing calculation allows us to compute 8.23 (a =2) times 24.3 (b =1) as follows 8.23 The intermediate calculations are as before. But we shift the decimal point in the product of 823 x 243 = 199989 three = 2 + 1 places to the left to obtain the result 199.989
Counting Digits in sums and difference of Decimal FractionsIf M and N are given by decimal fractions with
and a > b then
has a digits after the decimal point. |
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