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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. |
_______________________________________\
/
244 (read 4 ones plus 4 tens plus 2 hundreds ) |
- 31 (read 1 ones plus 3 tens plus 0 hundreds) |
---------- subtraction gives |
213 3 ones plus 1 tens plus 2 hundreds \|/
---------- | /____________________________________
\
365 (read 5 ones plus 6 tens plus 3 hundreds ) - 149 (read 9 ones plus 4 tens plus 1 hundreds ) ---------- ----------
10
3 6 5 (read 15 ones plus 5 tens plus 3 hundreds )
- 1 4 9 (read 9 ones plus 4 tens plus 1 hundreds )
---------- subtraction yields
6 1 2 6 ones plus 1 tens plus 2 hundreds
----------
-1 The -1 indicates a borrow and
The number above is obtained from
6 - 1 - 4
Second Example: Now consider 825 - 273
8 2 5 (read 8 hundreds plus 2 tens plus 5 ones )
- 2 7 3 (read 2 hundreds plus 7 tens plus 3 ones )
---------- This is the same as
8-1 hundreds plus 12 tens plus 5 ones
minus 2 hundreds plus 7 tens plus 3 ones. This yields
5 5 2 or 5 hundreds plus 5 tens plus 2 ones
----------
-1 This 1 below the bar indicates the conversion
of 8 hundred into
7 hundreds plus 10 tens -- the "borrow".
Third Example: Consider 8234 - 4816
Fourth Example: Another example (repeated borrows)
The pattern is as follows.
These three replacements imply 4823 equals 4-1 thousands + 18-1 hundreds +
12-1 tens plus 13 ones, Fifth Example - A Case of Repeated Borrows or Conversions
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Steps:
A more standard way to do this is to cross-out the 4000 and replace it by 3990 + 10 as follows.
(II) Column Method with Two Rows
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58.65
17.44 _
41.25
You will have 41.25 left.
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Imagine again that you have 5 ten dollar bills, 8 one dollar bills, 6 dimes and 5 pennies in a piggy bank. So again, the total amount in the piggy bank is 58.65 dollars. Suppose you owe another 29. 87 dollars. If you give 2 tens, 8 ones and 65 cents, you will have $ 30.00 left and still owe 1 one and 22 cents. The latter remains to take from the 30.00 -- we can write the following.
Here 7 from 5 pennies leaves 0 with 2 more owing or to subtract; 8 dimes from 6 dimes leaves 0 with 2 more owing; and 9 from 8 dollars leaves 0 with 1 more to be subtracted. To pay the debt completely, compute 30.00 - 1.22 as follows.
We can write all the foregoing at once:
So 58.65 = 29.87 = 28.78 Observe we subtract as much as we can in each column without borrowing (or converting). That gives two rows. The first row gives the amount that still remains. The second row shows what still needs to be subtracted. Examples follow.
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III: Column Method using ComplementsThere is another name for this that will return, or be found in one of my books. We introduce this complementary column method by solving for unknowns, and then re-arranging the rows in a way that hides the unknowns. Start With UnknownsFirst ExampleOne way to find or define 825 - 273 is to consider the missing number puzzle CBA The question here is what should the digits A, B and C equal given they belong to the set 0 to 9.
The foregoing gives C52 with a carry of 1 in the hundreds column. Now we find C so that the carry 1+ 2+ C = 8. By inspection, C = 5 Hence, we have or should have 552 That latter is easily checked by the column addition method.
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Now 825 = 552+ 273 . Therefore 825 -273 =
(552+273) -273 = 552.
The foregoing gives an alternative method for finding the
difference 825 - 273
Compute 8234 - 4816
Write
4816
DCBA
----- +
8234
-----
Want 6+A = 4 modulo 10. So A = 8 with a carry
of 1
DCB8
4816
----- +
8234
-----
1
Need 1+ 1 + B = 3 exactly or modulo 10. So B =
1 with no carry
Need 8+C = 2 exactly or modulo 10. So C = 4 with a carry of 1.
The foregoing gives
D418
4816
----- +
8234
-----
1 1
Now we need 1 + 4+ D = 8 exactly. So D = 3
5418
4816
----- +
8234
-----
1 1
Our conclusion is 5418 = 8234 - 4816.
By swapping the first and third row in the above calculations, we get a sequence of column method to do a subtraction via complements rather than borrows.
Write
8234
DCBA
----- -
4816
-----
Want 6+A = 4 modulo 10. So A = 8 with a carry
of 1
8234
4816
----- -
DCB8
-----
1
Need 1+ 1 + B = 3 exactly or modulo 10. So B =
1 with no carry
Need 8+C = 2 exactly or modulo 10. So C = 4 with a carry of 1.
The foregoing gives
8234
4816
----- -
D418
-----
1 1
Now we need 1 + 4+ D = 8 exactly. So D = 3
8234
4816
----- -
5418
-----
1 1
Our conclusion is 5418 = 8234 - 4816.
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Steps in the computation of 6855- 2985 follow - with letters 6855
6855
6855 5 + A = 5 modulo 10,
A = 0 |
|
Steps in the computation of 6855- 2985
follow - without letters 6855
6855
6855 5 + ? = 5 modulo 10,
? = 0 |
6855
2985
---- -
3870
-----
11
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Number TheoryStart of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions Number Theory
Continued
Decimal Place Value Comparison Method Addition Method Subtraction Methods Multiplication Methods Division Methods Remainder Arithmetic I Primes & Composites Primes Factorization Primes & Composites Prime Factorization Aids Prime Factorization Examples Counting Whole No. Factors Arithmetic Videos Square Roots Fractions & Decimals Fractions as Decimals 1 = 0.999 Recurring Long Division Continued Ratio of Simple Fractions Ratio of Decimal Fractions Unsigned Reals Numbers Signed Coordinates Plane Vectors Horizontal Vectors How to Add Reals How to Multiply Reals Distributive Law for Reals Remainder Arithmetic II
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