Let’s give it a quick review then I will jump into theory and derivation. Educators please note that
this page is a deviation from my Junior Elementary schoolers stuff. Though it does concern them
as have been demonstrated adequately in the pages above nevertheless, I will be delving into complexity that is beyond them and therefore this section could be skipped - at least for them until later on in their life.
15 x 27 can be written as ( 10 + 5 ) x ( 20 + 7 ) ,
expanding we get 10 x 20 + 10 x 7 + 20 x 5 + 7 x 5 .
Let’s jump ahead and come back a while later.
Note: There is really no reason for the multiplicands to be distributed around the rectangle as shown. The only thing to watch is that you don’t end up multiplying20 x 7 or 10 x 5. Have fun moving them around.
Now, compare the values in the cells to the expansion we just did above.
The are exactly the same. The rectangular matrix masked the expansion and
the Algebraic part of the operation.
The are exactly the same. The rectangular matrix masked the expansion and
the Algebraic part of the operation.
Generalization of the matrix
A three digits number can be written in the form a x 102 + b x 10 + c.To save space I will write the above as at2+ bt + c where t = 10
Therefore a 3 digits number x another 3 digits number can be expressed as:
(at2 + bt + c ) x ( xt2 + yt + z ) where a,b,c,x,y,z are numbers each of which
has value ranging from 0 to 9. Although, a and z can be 0 but if that
happens a and/or z will not be written. What I am saying is that we don’t
usually write 026 but 26 ( the 0 in this case is not required as a place holder.)
Any 2 of the 6 numbers when multiplied can have value ranging from 0 - 81.
Look at the one I draw for base value t2.
In fact, starting from cz which have base t0 (Unit value) each diagonal is 10 ten times
the previous. In other words, the diagonals move from t0 t1 t2 to t4. And since t represent 10 the
diagonals are marching in this order ( right to left ) TTh Th H T U which is exactly how we
represent numbers in the Decimal System.
Hence, cz represent the Unit value, cy + bz the Tens value etc.
There are five groups of numbers viz
• cz x 1
• (bz + cy ) x t
• (az + by + cx ) x t2
• (ay + xb) x t3
• ax x t4
Example 328 x 179 can be expressed in any of the following forms.
Both ways of representing the matrix have their respective merits and problems. The first one can cause confusion over the placement of the numbers - often leading to wrong answer. The second is not efficient because of all those 0(s) - however it does not cause confusion and the highest value in this case 30000 serve as the place holders - answer tends to be accurate.
Let’s look at the t2 numbers viz 2700 1400 and 800
Since t2 = 100 ,
2700 = 27 x 100 = 20 x 100 + 7 x 100
= 2 x 103 + 7 x 102 = 2000 + 700 = 2t3 + 7t2
All the 9 numbers in the matrix can be written in this form and should be.
If we apply this format, then the matrix above can be rewritten as
Let’s look at the t2 numbers viz 2700 1400 and 800
Since t2 = 100 ,
2700 = 27 x 100 = 20 x 100 + 7 x 100
= 2 x 103 + 7 x 102 = 2000 + 700 = 2t3 + 7t2
All the 9 numbers in the matrix can be written in this form and should be.
If we apply this format, then the matrix above can be rewritten as
And then finally this.
Since the diagonals implicitly represent the base values , we
can drop the zeros from the nine numbers.
Let’s try some more examples
can drop the zeros from the nine numbers.
Let’s try some more examples
It feels strange to me having one of the multiplicands
on the right side as shown. However, it has its advantage.
The digits of the answer are aligned perfectly with the
diagonals - answer : 666225
This matrix I am using is known as Chinese Lattice Method of Multiplication.
Using the Rectangle matrix requires a minimum number of squares.
2 x 2 requires 4 squares
3 x 3 requires 9 squares
The number of squares required increases rapidly with the increase in digits
in the multiplicands and can cause this method to be unwieldy.
Like many tricks and methods of the past this one will soon be abandoned
as the schoolers grow in age - smart phones will make all these obsolete.
Actually I derived this independently of the Chinese and I am doubtful if
the Chinese were the first as the Mathematics behind the derivation is actually
simple. However the observation that the lattice structure align all the values
with the same base value is what I would like to commend them on.
on the right side as shown. However, it has its advantage.
The digits of the answer are aligned perfectly with the
diagonals - answer : 666225
This matrix I am using is known as Chinese Lattice Method of Multiplication.
Using the Rectangle matrix requires a minimum number of squares.
2 x 2 requires 4 squares
3 x 3 requires 9 squares
The number of squares required increases rapidly with the increase in digits
in the multiplicands and can cause this method to be unwieldy.
Like many tricks and methods of the past this one will soon be abandoned
as the schoolers grow in age - smart phones will make all these obsolete.
Actually I derived this independently of the Chinese and I am doubtful if
the Chinese were the first as the Mathematics behind the derivation is actually
simple. However the observation that the lattice structure align all the values
with the same base value is what I would like to commend them on.
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