Multiplication tables and proportion tables

Here is a typical modern multiplication table. (Older tables usually go up to `12 xx 12`.)

`\times`	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

At the left and along the top are the stubs---here, in this example, the stubs are colored. The rest is the body of the table. One looks up the numbers to be multiplied in the stubs, and then finds the product in the body, at the intersection of an appropriate row and column. So, for instance, `"4 "xx" 5 = 20"`:

`\times`	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

The standard multiplication table and how to use it are presumably familiar.  We can put arbitrary quantities in the stubs, and fill out the body of the corresponding multiplication table :

`\times`	`2 1/2`	`4`	`1`	`600`	`7" hours"`
`2`	`5`	`8`	`2`	`1200`	`14" hours"`
`4" mph"`	`10" mph"`	`16" mph"`	`4" mph"`	`2400" mph"`	`28" miles"`
`1`	`2 1/2`	`4`	`1`	`600`	`7" hours"`
`9`	`22 1/2`	`36`	`9`	`5400`	`63" hours"`
`100`	`250`	`400`	`100`	`60000`	`700" hours"`
`8`	`20`	`32`	`8`	`4800`	`56" hours"`
`1/3`	`5/6`	`1 1/3`	`1/3`	`200`	`2 1/3" hours"`

One can build a multiplication table with any number of rows and any number of columns.

In the body of a multiplication table :

• every row is proportional to every other row, and
• every column is proportional to every other column.

These two properties are linked.  Any table of numbers that has one of these properties also has the other.  Let's call this property the proportionality condition. 

Given a single nonzero row, one can reconstruct any other row from the value at a single position in the row to be reconstructed, and mutatis mutandis for columns.

Any table of numbers that can be the body multiplication table is called a proportion table.  To be a proportion table, then, a table of numbers has only :

• satisfy the proportionality condition on its rows/columns.

The name proportion table is justified not just by the proportionality of row to row and column to column, but also by the connection to proportion problems of the kind once solved by the Rule of Three. 

The following are equivalent:

Old proportion notation:
`\qquad\qquad 750" men : "22,500" rations of bread :: "1200" men : What?"`

Algebraic notation:
`\qquad\qquad\qquad\frac{750" men"}{"22,500 rations of bread"}=\frac{1200 " men"}{x}.`

Proportion table form:

`"750 men"`	`"1200 men"`
`"22,500 rations of bread       "`	`"What?"`

To find a missing entry in a `2 \times 2` proportion table, one multiplies the neighboring entries, and divides by the opposite entry:

`\qquad\qquad\qquad x  = \frac{"22,500 rations of bread" \times "1200 men"}{"750 men"},`

`\qquad\qquad\qquad    = "36,000 rations of bread".`

This three argument operation---multiplying two numbers and dividing by a third---is remarkably useful. One computer language, FORTH, even defines a single operation, '*/' or starslash, for doing this.

We can always find a solution as long as we have three appropriately positioned values, and as long as we don't have to divide by zero. A zero in the top stub of the multiplication table leads to a column of zeroes in the body of the table; a zero in the side stub leads to a row of zeroes in the body. A zero only appears in the body of the table as part of a row stub or column stub. In short, a zero appears somewhere in the stubs exactly when a zero appears in the body of the table. A table with no zeroes is called zero free. We shall prefer to work with zero free proportion tables.

In writing this proportion table calculation, we have followed the convention of writing the row neighbor before the column neighbor in the numerator, with the opposite centered below them in the denominator. Long ahead, when we get to group tables, this convention will prove useful.