Why is the standard multiplication table harder to learn than the usual addition tables?

A standard times table, in proportion table form, looks like this :

`\qquad\qquad\qquad[(1,2,3,4,5,6,7,8,9,10),(2,4,6,8,10,12,14,16,18,20),(3,6,9,12,15,18,21,24,27,30),(4,8,12,16,20,24,28,32,36,40),(5,10,15,20,25,30,35,40,45,50),(6,12,18,24,30,36,42,48,54,60),(7,14,21,28,35,42,49,56,63,70),(8,16,24,32,40,48,56,64,72,80),(9,18,27,36,45,54,63,72,81,90),(10,20,30,40,50,60,70,80,90,100)]`
 
A standard addition table (not a proportion table of course, but rather a glide table,) looks like this :

`\qquad\qquad\qquad[(0,1,2,3,4,5,6,7,8,9),(1,2,3,4,5,6,7,8,9,10),(2,3,4,5,6,7,8,9,10,11),(3,4,5,6,7,8,9,10,11,12),(4,5,6,7,8,9,10,11,12,13),(5,6,7,8,9,10,11,12,13,14),(6,7,8,9,10,11,12,13,14,15),(7,8,9,10,11,12,13,14,15,16),(8,9,10,11,12,13,14,15,16,17),(9,10,11,12,13,14,15,16,17,18)]`

A glide table is to addition and subtraction what a proportion table is to multiplication and division.

In a glide table, all the rows, and equivalently all the columns, differ by fixed amounts.

The way to reconstruct a value at one corner of a box (`2 xx 2` subtable) also differs, but in an obvious way :  to find the value at one corner, one adds the values at the neighboring corners and subtracts the value at the opposite corner.

So now we have presented two tables, one for multiplication, and the other for addition.  It is reasonably common for people to not remember all multiplications in the times table, but rather rarer for them not to know all that is in the addition table.

There are a number of reasons for this, of course.  The numbers in the multiplication table are mostly larger, for instance.

Two of the reasons, however, for the relative ease of learning the addition table can help us construct multiplication tables that are easy, for what they do:

• the standard addition table, even though it is the same size, has fewer distinct values than the standard, multiplication table, and
• the standard addition table has a simpler structure---it is stripy.

Looking at the standard addition table, we see stripes of common values.  It is possible to make multiplication tables that are stripy.  Consider, for instance, this one :

`\qquad\qquad\qquad[(1,2,4,8,16,32),(2,4,8,16,32,64),(4,8,16,32,64,128),(8,16,32,64,128,256),(16,32,64,128,256,512),(32,64,128,256,512,1024)]`

In the next posting, we shall begin constructing some useful stripy multiplication tables.