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By: Steven Kowalski
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Kowalski Box Method (KBM) vs. Regular MethodA preview
How this method was envisioned Trick to multiplying AB*11
Digits Normal Digit Notation Comma Notation
Kowalski Box Method 2-digit by 1-digit 2-digit by 2-digit 3-digit by 2-digit
General formula: n-digit by m-digit
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142
x171
294+420
714
42
x17
4,3|0,1|47 1 4
More examplestoward the end
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Let AB be a 2-digit number such as 2323
x112 5 3
(no need to write
2a and 2b)
2 + 3
Carry the 1 if necessary.
1
2a
1
2a
2b
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Fancy trick!I know, right?
Could we do this with other numbersbesides 2-digit numbers multiplied by
11?
How would you do it given any two Wholenumbers?
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TheArabic Numeralsare the figures{0,1,2,3,4,5,6,7,8,9}, each with anumerical
value These are the most common numerals used to
construct numbers.
Each number is composed ofdigits,numbers with a certainplace value. ones-place, tens-place, hundreds-place, etc.
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Let A, B, C, D, E be any Arabic numeralsand let ABCDE be the numeral string.
Then, ABCDE = A*10
4
+ B*103
+ C*102
+D*101 + E*100
Ex: A=1, B=5, C=3, D=4, E=8. Then ABCDE =15348
ABCDE = 1*10000 + 5*1000 + 3*100 + 4*10 +8*1ABCDE = 1*104 + 5*103 + 3*102 + 4*101 + 8*100
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ABCDE = 1 5 3 4 8
Let X be a number with n place values,each of which contains a numeral
X = Xn Xn-1 X3 X2 X1
Ones
place
Tens
place
Hundreds
place
Thousands
place
Ten-
thousandsplace
100s
place
101s
place
102s
place
10n-2s
place
10n-1s
place
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Suppose we want to write a number with ANYnumber in the place values (not just the numerals).Then you use a comma to separate each place value.
So, given ABCDE = 15348, it can be expressed asfollows:
= 1,5,3,4,8 = 1*104 + 5*103 + 3*102 + 4*101 + 8*100 = 15,3,4,8 = 15*103 + 3*102 + 4*101 + 8*100 = 1,5,0,34,8 = 1*104 + 5*103 + 0*102 + 34*101 + 8*100 = 1,4,13,0,48 = 1*104 + 4*103 + 13*102 + 0*101 + 48*100 = 153,1,38 = 153*102 + 1*101 + 38*100 (Diagram on next page)
So, in using comma notation to write a number, wesee that in between each comma can be ANY non-negative integer (or Whole number).
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15348 = 1 , 4 , 13 , 0 , 48
15348 = 153 , 1 , 38
[100s]
Onesplace
[101s]
Tensplace
[102s]
Hundredsplace
[103s]
Thousandsplace
[104s] Ten-
thousandsplace
Ones
place
Tens
place
Hundreds
place
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In general, where X{1,2,3,,n-1,n} are non-negative integers, if X is written in commanotation as:
X = Xn, Xn-1,, X3, X2, X1Then
X = Xn*10n-1+Xn-1*10n-2++X2*101+X1*100And this can be
expressed by:
But there has to be an easier way tocalculate it!!!!!!!!
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The Box Method: Given a number in comma notation, suchas X = 12,315,84 ,
1. Draw a line before the ones-place digit for each place inthe comma notation: 12, 315, and 84
1|2,31|5,8|42. Add the numbers in each box (from right to left), carrying
remainders to the next compartment. 1|2,31|5,8|4
1|2+31|5+8|4 (commas mean plus)
1+3|3+1 | 3 |4 (added right to left, carrying remainders)
4 4 3 4 (added right to left, no remainders to carry)
More Condensed 1+3|2,31+1|5,8|4
4 4 3 4 (added right to left carrying each remainder,
while dropping the ones place)
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This method is useful especially whenmultiplying 3-digit numbers by 3-digit
numbers or less.The multiplication can be done from right
to left or from left to right.
It also takes up less space on paper.
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Let AB and C be
whole numbers:
AB*C =
(10A*C+1B)*(1C)
= 10A*C+1B*C
= A*C,B*C
AB
xC
A*C,B*C
25
x6 .
1|2,3|01 5 0
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Let AB and CD be wholenumbers:
AB*CD=
(10A+1B)*(10C+1D)=100A*C+10A*D+10B*C
+1B*D=100(A*C)+10(A*D
+B*C)+1(B*D)
ABxCD
A*C,A*D+B*C,B*D
25x61 .
1|2,3|2,|51 5 2 5
Or25
x 61 .
1|2,3|0,|5| |2 | .1 5 2 5
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Let ABC and DE bewhole numbers:
(100A+10B+1C)*(10D+1E)
=1000A*D+100A*E+100B*D+10B*E+10C*D+1C*E
=1000A*D+100(A*E+B*D)+10(B*E+C*D) +1C*E
ABCx DE
A*D,A*E+B*D,B*E+C*D,C*E
514x42 .
2|0,1|4,1|8,|8 .2 1 5 8 8
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Let X = Xn*10n-1+Xn-1*10
n-2++X2*101+X1*10
0 and
Y = Yp*10p-1+Yp-1*10
p-2++Y2*101+Y1*10
0where npThen
But Lets make this one summation:
Did it backwards but dont want to rewrite it Sorry
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Ill get back to it tomorrow (promise)because Im too tired to keep writing.
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