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Visual Aids and Column Multiplication Methods

The association of products of whole numbers with counting sub-rectanglar divisions of a larger rectangle leads to visual aids for developing and remembering the generalized distributive law for whole numbers, fractions, proper or not, and nonnegative real numbers. 

Remark: For signed numbers and polynomials with negative coefficients we  may replace the visual aids by column multiplication methods. Details follow later. 

Go to the site Number theory areas to learn more when you have time to spare

First Example (Alternative to Foil Method)

How do we express a product 

NM = (a+b)(c+d) 

as a expression of the terms a, b, c and d?

Solution:  The number NM gives the number of subrectangles in the blue rectangle below.

The BLUE rectangle can be divided into  4 intermediate size subrectangles

Each intermediate rectangle labelled I to VI contains a number of the NM subrectangles we are counting. Each of the NM subrectangles we are counting belongs to one and only one of the intermediate size rectangles. See below.

Column Multiplication Method

We have use the product rule for counting subrectangles to find the number of subrectangles of the total MN in each intermediate subrectangle. The intermediate size rectangles lead to four groups of subrectangles with counts ac, ad, bc and bd we can be add to obtain the total number MN. 

So  MN= (a+b)(c+d) = ab+ad + bc + bd.

We may introduce a column multiplication method to obtain the product 

 c + d
 a + b                  _x 
ac + ad                   = product of first row with a
bc + bd              ₊   = product of first row with b
ab + ad + bc + bd   =  (a+b)(c+d)

Here ab+ad + bc + bd  and   (a+b)(c+d) give two different ways to compute a single number, the number of subrectangles MN.  The equality of two different ways to compute a single number gives many formulas in mathematics. 

Second Example 

How do we express a product 

NM = (a+b+c)(e+f) 

as a expression of the terms a to f giving each factor.

Solution:  The number NM gives the number of subrectangles in the blue rectangle below.

The BLUE rectangle can be divided into  6 intermediate size subrectangles

Each intermediate rectangle labelled I to VI contains a number of the NM subrectangles we are counting. Each of the NM subrectangles we are counting belongs to one and only one of the intermediate size rectangles. See below.

We have use the product rule for counting subrectangles to find the number of subrectangles of the total MN in each intermediate subrectangle. The intermediate size rectangles lead to six groups of subrectangles with counts ea, eb, ec, fa, fb and fe we can be added in any order to obtain the total number MN. 

So  MN= (a+b+c)(e+f) = ea + eb + ec+ fa+ fb +fc 

We may introduce a column multiplication method to obtain the above product

Remark 1: The foregoing visual or geometric derivation the generalized distributive law  holds for non-negative rational and irrational numbers a to f with unit length in place of the word rows and columns if we derive and then use the additive properties of area - the area of a rectangle equals the sum of areas of a set of subrectangles covering it - subrectangles which intersect only at their edges. Details will be given later. 

Column Methods for Multiplication

 We may replace the rectangles above by multiplication tables in which the terms in the factors provide the initial entries in rows and columns. 

Further table entries are obtained via products. The foregoing can be tabulated as a column method for multiplication:

 a + b + c
 e + f                           _x 
ea + eb + ec                 = product of first row with e
fa + fb + fc                   ₊   = product of first row with f
ea + eb + ec+ fa+ fb +fc  =   (e+f)(a+b+c) or (a+b+c)(e+f) when multiplication
                                       is commutative

Third Example - Product of Polynomials

Use rectangles to expand the product 

P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)

where b is a whole number. 

Solution:  Form intermediate size rectangles 

and compute the number of subrectangles in each. The latter operation first gives

and then this 

The foregoing could have been done in one step. Now instead of add the intermediate rows by rows, we will add them along the diagonals in the powers of b are identical. 

If we have made no mistakes, the foregoing gives the result

P =  300  + (300 + 48)b + (200 + 36 + 32)b²  + (500+24+24)b³  +  (60+16)b⁴  +  40b⁵ 

P = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵ 

Whether or not the powers of b increase or decrease in the  result is a cosmetic convention - some prefer one way, others the other, and some either. 

Our conclusion follows. The product

(50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)  = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵ 

That is computation of the left hand side for a given value of b gives the same result as computation of the right. 

Multiplication Table Approach.

The above geometric approach suggests a table method:

which holds for real number b as well as whole numbers since the generalized distributive law holds for real numbers as well as whole numbers (why to come later). Here the rows and columns of the table come from the terms in the factors of the product to be computed, here (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³) while the entries inside list or tabulate the products of pairs of terms, one from each factor.

Column Method for Multiplication of Polynomials

Here we modify the table approach and dedicate a column to each power of b as follows

In this modified table approach, we compute all possible products as before, but align the products in each row according to their power of b. That makes addition and collecting like powers of b (with the aid of the distributive law) simpler. We add an extra row for the sum.

So the product P = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵   as before

In retrospect, the table (and column multiplication below) will be easier to do if we arrange the powers of b in ascending (that is, increasing) order along the top row.

and align the left column at the left of the page (or page margin), so that we may write from left to right. Here again all possible products appear, but aligned in columns. 

Finally, let us introduce or switch to the table method for multiplication. 

Column Method for Products of Polynomials

8   +      6 b +   4b²  +  10b³ 
50 +      6b   +  4b²           _x
300 + 300b + 200b² + 500b³                              (product of top row with 50)
             48b  +  36b²  +   24b³  +  60b⁴                 (product of first row with 6b)
                        32b²  +   24b³   + 16b⁴   + 40b⁵  (product of first row with 4b²)
^{------------------------------------------------------------------------}
300 + 348b + 268b²  + 548b³  +  76b⁴  +  40b^{5 
------------------------------------------------------------------------}

Note: The ascending order appears to work best as the position of the  intermediate products shifts to the right away from the left margin where the computation begins.  Do you see how each entry in each row of the column method corresponds to an entry in the rectangular approach and the two preceding tabular approaches to the computation of the product? All is a consequence of the distributive seen algebraically or geometrically visualized.

The case where the variable b is replaced by the number 10 leads to a justification of the column method for multiplication of whole numbers using their decimal representation with powers of 10 written in decreasing order rather than increasing.  

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science