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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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Decimals Multiplication RevisitedThe first six of number theory lessons
provide a logical framework for this page. But you may explore this page first if you like. The decimal representation of a whole number can be viewed as polynomial in powers of 10 with coefficients limited to the digits 0 to 9 except during computations. See below. Here 243 = 2 x 102+ 4 x 10 + 3 and 823 = 8 x 102+ 2 x 10 + 3. To compute the product
of these two numbers, we form the rectangle and imagine we are counting subrectangles due to the intersection of 243 rows and 843 columns.
Multiplication yields 6 intermediate rectangles with the indicated number of subrectangles.
So addition along diagonals to group like powers of 10 gives
All the foregoing with some cosmetic rearrangement justifies column methods for multiplication of whole numbers using their decimal representation. And in the column method, the conversion and simplification are done as part of the computation and not after. 8 x 102+ 2 x 10 + 3 = 823 In compact decimal notation, with carries and so on, we may rewrite the foregoing as 823 Observe how the expanded form of the calculations with polynomials in powers of 10 leads to and justifies the standard column method for multiplication of whole numbers using their decimal representation. Decimal Methods for Multiplication - more examples in compact notation.Just as there were carries in addition, there are carries in multiplication. Example. Consider 3 times 451 = 4 hundred + 5 tens + 1 one. The answer is 12 hundreds + 15 tens + 3 ones or 12+1 hundreds + 5 tens + 3 ones. In shorthand form, we may write 451 451 451
x 3 or x 3 or using carries x 3
---- --- ----
1200 3 1353
150 150 ----
+ 3 1200 1
------ ----- a more condensed
1353 1353 or compact form.
----- ----
Similar Example. Consider 7 times 452.
452 452 452
x 7 or x 7 or using carries x 7
---- --- ----
2800 14 3164
350 350 ----
+ 14 2800 31 <-- the carries
------ -----
3164 3164 (Sometimes the carries are
----- ---- not written --> less writing.
Third Example. Consider 25 times 3438 = (20+ 5) times 3438. 3438 x 25 ------ 17190 <----- Times 5 68760 <----- Times 20 --------- + 75950 <----- Times 20+5 ---------- Observation: If the whole number factor have a and b digits respectively in their decimal representation then their product has a+b digits. That is analogyous to multiplication of polynomials. The |
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