Stripy multiplication tables

Consider the following (somewhat contrived) stripy multiplication table:

`\qquad\qquad\qquad[(1,100, 10000),(100,10000,1000000),(10000,1000000,100000000)]`

It is easy to interleave new rows and new columns, to get another stripy multiplication table :

`\qquad[(1,10,100,1000,10000),(10,100,1000,10000,100000),(100,1000,10000,100000,1000000),(1000,10000,100000,1000000,10000000),(10000,100000,1000000,10000000,100000000)]`

Now look at this very simple table, which is a subtable of the last one :

`\qquad\qquad\qquad[(1,10),(10,100)]`.

Can we interleave a new row and a new column into this table, preserving the existing values, and producing a new stripy multiplication table? In other words, can we fill in the missing values in this,

`\qquad\qquad\qquad[(1,"*",10),("*","*","*"),(10,"*",100)]`,

to make the result a stripy proportion table? Stripiness is constancy along the diagonals :

`\qquad\qquad\qquad[(1,u,10),(u,10,v),(10,v,100)]`.

The proportionality condition requires e.g. that

`\qquad\qquad\qquad1/u = u/10,`

i.e. that `u^2 = 10,` i.e. that `u = \pm\sqrt10`. Let us choose `u = \sqrt{10}.` Then we have

`\qquad\qquad\qquad[(1,\sqrt10,10),(\sqrt10,10,v),(10,v,100)]`.

Looking at the subtable in the top right hand corner,

`\qquad\qquad\qquad[(\sqrt10,10),(10,v)]`,

we can solve for `v` :

`\qquad\qquad\qquad v = \frac{10 xx 10}{\sqrt10},`
`\qquad\qquad\qquad\qquad = \frac{100}{\sqrt10},`
`\qquad\qquad\qquad\qquad = \frac{100 xx \sqrt10}{\sqrt10 xx \sqrt10},`
`\qquad\qquad\qquad\qquad = \frac{100 xx \sqrt10}{10},`
`\qquad\qquad\qquad\qquad = 10\sqrt10.`

So now we have a complete stripy multiplication table :

`\qquad\qquad\qquad[(1,\sqrt10,10),(\sqrt10,10,10\sqrt10),(10,10\sqrt10,100)]`.

This can be more revealingly presented by writing all the entries as powers of `10`:

`\qquad\qquad\qquad[(10^0,10^{1/2},10^1),(10^{1/2},10^1,10^{1 1/2}),(10^1,10^{1 1/2},10^2)]`,

or if we write the exponents fully in decimal,

`\qquad\qquad\qquad[(10^0,10^{0.5},10^1),(10^{0.5},10^1,10^{1.5}),(10^1,10^{1.5},10^2)]`.

This illustrates a general truth. A stripy proportion table is made by raising a single number (i.e. a base) to powers (exponents) coming from a stripy glide table. Here, for instance, we can pick `10` as the base, and

`\qquad\qquad\qquad[(0,0.5,1),(0.5,1,1.5),(1,1.5,2)]`

as the table of exponents.

The choice is not unique, however.  For instance, we could get the same result by using `\sqrt10` or `10^{0.5}` for the base, and

`\qquad\qquad\qquad[(0,1,2),(1,2,3),(2,3,4)]`

for the table of exponents.  Either way, we the result is the multiplication table

`\qquad\qquad\qquad[(10^0,10^{0.5},10^1),(10^{0.5},10^1,10^{1.5}),(10^1,10^{1.5},10^2)]`.

Let us rewrite this in decimals, rounding to three significant figures :

`\qquad\qquad\qquad[(1,3.16,10),(3.16,10,31.6),(10,31.6,100)]`.

When working with stripy multiplication tables, lets us adopt the understanding that rounded values are shorthand for the exact values.

We can fill in new rows between these.  Stripiness constrains what is possible :

`\qquad\qquad\qquad[(1,a,3.16,b,10),(a,3.16,b,10,c),(3.16,b,10,c,31.6),(b,10,c,31.6,d),(10,c,31.6,d,100)]`.

Consider, now, just the first two columns :

`\qquad\qquad\qquad[(1,a),(a,3.16),(3.16,b),(b,10),(10,c)]`.

By stripiness, the second column is essentially the first column, displaced upward one place.  Applying the proportionality condition to the first two rows, we have :

`\qquad\qquad\qquad1/a = a/3.16`.

Then `a^2 = 3.16` and `a = \pm1.78`, where we are using `3.16` as shorthand for `\sqrt{10}` and `1.78` as shorthand for `\sqrt\sqrt10`.  Selecting the positive sign, `a = 1.78`.

Now, considering the second and third rows, we have

`\qquad\qquad\qquad1.78/3.16 = 3.16/b`.

This determines `b` fully, including its sign.  Alternatively, we can look at the third and fourth rows, taking

`\qquad\qquad\qquad3.16/b = b/10`,

and this determines that the magnitude of `b` is `\sqrt{3.16 xx 10} = 5.62`, i.e. that it is the geometric mean of `3.16` and `10`.

Looking back, we see that with the choice of positive sign, `a` is also the geometric mean of the `1` and `3.16`, the quantities above and below it.

`\qquad\qquad\qquad[(1,1.78),(1.78,3.16),(3.16,5.62),(5.62,10),(10,c)]`.

The last value can be conveniently found from applying the proportionality condition to the first and last rows, i.e. to the four corners of the preceding table,

`\qquad\qquad\qquad1/10 = 1.78/c`,

so that `c = 10 xx 1.78 = 17.8`, and the table is

`\qquad\qquad\qquad[(1,1.78),(1.78,3.16),(3.16,5.62),(5.62,10),(10,17.8)]`.

By stripiness, we can generate most of the third column,

`\qquad\qquad\qquad[(1,1.78,3.16),(1.78,3.16,5.62),(3.16,5.62,10),(5.62,10,17.8),(10,17.8,"d")]`.

The proportionality condition on the four corners of this table is

`\qquad\qquad\qquad1/10 = 3.16/d`,

whence `d = 10 xx 3.16 = 31.6,` and the table becomes

`\qquad\qquad\qquad[(1,1.78,3.16),(1.78,3.16,5.62),(3.16,5.62,10),(5.62,10,17.8),(10,17.8,31.6)]`.

We can find the remaining columns in the same way, getting, to three significant digits,

`\qquad\qquad\qquad[(1,1.78,3.16,5.62,10),(1.78,3.16,5.62,10,17.8),(3.16,5.62,10,17.8,31.6),(5.62,10,17.8,31.6,56.2),(10,17.8,31.6,56.2,100)]`.

As before, we can express this as

`\qquad\qquad\qquad[(10^0,10^0.25,10^0.5,10^0.75,10^1),(10^0.25,10^0.5,10^0.75,10^1,10^1.25),(10^0.5,10^0.75,10^1,10^1.25,10^1.5),(10^0.75,10^1,10^1.25,10^1.5,10^1.75),(10^1,10^1.25,10^1.5,10^1.75,10^2)]`.

i.e, as `10` to the power of exponents from the stripy glide table

`\qquad\qquad\qquad[(0,0.25,0.5,0.75,1),(0.25,0.5,0.75,1,1.25),(0.5,0.75,1,1.25,1.5),(0.75,1,1.25,1.5,1.75),(1,1.25,1.5,1.75,2)]`.

Using stripiness and proportionality, the proportion table

`\qquad\qquad\qquad[(1,1.78,3.16,5.62,10),(1.78,3.16,5.62,10,17.8),(3.16,5.62,10,17.8,31.6),(5.62,10,17.8,31.6,56.2),(10,17.8,31.6,56.2,100)]`

can be extended infinitely in all directions. Here's a start :

`\qquad[(0.1,0.178,0.316,0.562,1,1.78,3.16,5.62,10),(0.178,0.316,0.562,1,1.78,3.16,5.62,10,17.8),(0.316,0.562,1,1.78,3.16,5.62,10,17.8,31.6),(0.562,1,1.78,3.16,5.62,10,17.8,31.6,56.2),(1,1.78,3.16,5.62,10,17.8,31.6,56.2,100),(1.78,3.16,5.62,10,17.8,31.6,56.2,100,178.),(3.16,5.62,10,17.8,31.6,56.2,100,178.,316.),(5.62,10,17.8,31.6,56.2,100,178.,316.,562.),(10,17.8,31.6,56.2,100,178.,316.,562.,1000)]`.

This infinite multiplication table can be built from very little.

First, we note that every row and every column consists of the same infinite-in-both-directions sequence of values,

`\qquad\ldots,0.316,0.562,1,1.78,3.16,5.62,10,17.8,31.6,\ldots`

Second, we note that this is what is classically called a geometric progression, a sequence of numbers with a common ratio.  Each number is `10^0.25 \approx 1.78` times the one that precedes it.  Since `1` is in this sequence, the sequence consists of all numbers `10^{n/4}`, with `n` an integer.

Third, we note that, up to round factors of ten, there are only four distinct numbers here, even in the infinite version of the table.  To three significant digits, they are : `1, 1.78, 3.16, 5.62`.  Every number in the table consists of one of these digit strings, differing only where the decimal point is placed.

To have this infinite table, to three digit accuracy, at one's mental service, all one need do is remember this sequence of numbers and what they mean :

`\qquad\qquad\qquad[(\ulx,\ul{10^x}),(0,1),(1//4,1.78),(1//2,3.16),(3//4,5.62),(1,10)]`.

To find, for instance, `56.2 xx 3.16`, one reasons

`\qquad\qquad\qquad 56.2 xx 3.16 = 10 xx 5.62 xx 3.16,`
`\qquad\qquad\qquad\qquad = 10^1 xx 10^\frac{3}{4} xx 10^\frac{1}{2},`
`\qquad\qquad\qquad\qquad = 10^(1 + 3/4 + 1/2),`
`\qquad\qquad\qquad\qquad = 10^(2 + 1/4),`
`\qquad\qquad\qquad\qquad = 10^2 xx 10^{1/4},`
`\qquad\qquad\qquad\qquad = 100 xx 1.78,`
`\qquad\qquad\qquad\qquad = 178.`

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.