Kowalski Box Method

8/3/2019 Kowalski Box Method

1/18

By: Steven Kowalski


2/18

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


3/18

142

x171

294+420

714

42

x17

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 write

2a 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 numbersbesides 2-digit numbers multiplied by

11?

How would you do it given any two Wholenumbers?


6/18

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.


7/18

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


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

100s

place

101s

place

102s

place

10n-2s

place

10n-1s

place


9/18

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).


10/18

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


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*10n-2++X2*101+X1*100And this can be

expressed by:

But there has to be an easier way tocalculate it!!!!!!!!


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 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)


13/18

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.


14/18

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


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 5

Or25

x 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+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


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

0where 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

Documents

Transcript of Kowalski Box Method