kowalski box method (2)

8/3/2019 Kowalski Box Method (2)

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By: Steven Kowalski


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Kowalski Box Method (KBM) vs. Regular Method A 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


3/18

142x17

1294+420714

42x17

4,3|0,1|47 1 4

More examplestoward the end


4/18

Let AB be a 2-digit number such as 2323

x112 5 3

(no need to write2a and 2b)

2 + 3

Carry the 1 if necessary.

1

2a

1

2a

2b


5/18

Fancy trick!I know, right?Could we do this with other numbers

besides 2-digit numbers multiplied by 11?How would you do it given any two Whole

numbers?


6/18

The Arabic Numeralsare the figures{0,1,2,3,4,5,6,7,8,9}, each with a numericalvalue These are the most common numerals used to

construct numbers.

Each number is composed of digits,

numbers with a certain place value. ones-place, tens-place, hundreds-place, etc.


7/18

Let A, B, C, D, E be any Arabic numeralsand let ABCDE be the numeral string.

Then, ABCDE = A*104 + 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*1 ABCDE = 1*104 + 5*103 + 3*102 + 4*101 + 8*100


8/18

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

100splace

101splace

102splace

10n-2splace

10n-1splace


9/18

Suppose we want to write a number with ANY numberin the place values (not just the numerals). Then youuse 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, we see

that in between each comma can be ANY non-negativeinteger (or Whole number).


10/18

15348 = 1 , 4 , 13 , 0 , 48

15348 = 153 , 1 , 38

[100s]Ones

place

[101s]Tens

place

[102s]Hundreds

place

[103s]Thousands

place

[104s] Ten-thousands

place

Ones

placeTens

placeHundreds

place


11/18

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*10

n-2++X2*101+X1*10

0

And this can be

expressed by:

But there has to be an easier way to calculateit!!!!!!!!


12/18

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 in thecomma notation: 12, 315, and 84 1|2,31|5,8|4

2. Add the numbers in each box (from right to left), carryingremainders 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)


13/18

This method is useful especially whenmultiplying 3-digit numbers by 3-digitnumbers or less.

The multiplication can be done from rightto left or from left to right.

It also takes up less space on paper.


14/18

Let AB and C bewhole numbers:

AB*C =

(10A*C+1B)*(1C)= 10A*C+1B*C

= A*C,B*C

ABxC

A*C,B*C

25x6 .

1|2,3|01 5 0


15/18

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 5Or

25x 61 .

1|2,3|0,|5| |2 | .1 5 2 5


16/18

Let ABC and DE bewhole numbers:

(100A+10B+1C)

*(10D+1E)=1000A*D+100A*E+1

00B*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


17/18

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

0 where npThen

But Lets make this one summation:

Did it backwards but dont want to rewrite it Sorry


18/18

Ill get back to it tomorrow (promise)

because Im too tired to keep writing.

kowalski box method (2)

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