11. Multiplying

When we were looking at the multiplication tables we were adding a constant, in the case of the 2 times table we stated with a 2 then added 2 to make 4 then we would add 2 again to make 6 and so on.

In this section we are going to be multiplying by a constant, not adding. This makes some interesting patterns one of which we’ve seen before.

Multiplying by 1
Anything we multiply by 1 will always create an exact copy of itself. Let start a sequence with 1 then multiply that by 1 equals 1 multiplying that by 1 would gives us 1. If we start with a 2 we are only ever going to get 2s. So multiplying by 1 doesn’t do that much.

Multiplying by 2
We have seen this pattern already expanding out of the multiplication ring set.
Lets start with a 1 now we multiply that by 2 and we get 2, so our sequence starts 1 2 we multiply the 2 by 2 equals 4, multiply by 2 equals 8, multiply by 2 equals 16 which implodes to 7 multiply by 2 equals 14 which implodes to 5 multiply by 2 equals 10 which implodes to 1 and if we continue we start the whole process again.

This 6 digit number sequence is one of the most important.
1 2 4 8 7 5
It is often refereed to as doubling and is a part of the Marko Rodin symbol.

00rodinssymbol

If you start at the 1 and draw a line to the 2 and continue through the rest of the sequence you create this symbol.
Its interesting to note that if you take any of the multiplication sequences and draw a line following this doubling sequence you get the same symbol.

21

The doubling sequence is missing the 3 6 and 9 so lets start a new sequence starting with a 3, we multiply by 2 to get 6 if we multiply 6 by 2 we get 12 which implodes to 3. So doubling just goes back and forth between the 3 and the 6.

The 1 2 4 8 7 5 is the finite female pattern of the infinite doubling male pattern and because it repeats infinitely we can draw it in a circle then expand it out to create a ring set.

x2_x5

This is a closed ring set, we can’t expand out to the next ring as we will only create another ring of 9s, we can still go around each ring multiplying by 2.

Another point is that 1 2 4 8 7 5 is the only order these digit could be arranged that would create the ring of 3 6 3 6 3 6.

The multiplication ring set we made had this doubling up pattern 1 2 4 8 7 5 as we expanded through each ring but it also had the 3 6 pattern if we look at the family member groups we had the 3 group on the first ring then the 6 group on the second and so on.

Multiplying by 3
If we start with any digit and multiply by 3 we do not create any repeating patterns, it just crashes into 9. Lets have a look
1 3 9
2 6 9
3 9
4 3 9
5 6 9
6 9
7 3 9
8 6 9
If we continue any of the sequences they would just keep creating 9s.

Multiplying by 4
When multiplying by 4 we create family member groups, lets start with a 1, multiply by 4 equals 4 multiply 4 equals 16 which implodes to a 7 multiply 4 equals 28 which implodes back to a 1.
1 4 7
If we do the same starting with a 2 we create the 6 family group.
2 5 8
Now multiplying any member of the 9 family group by 4 has no effect on the female number
3 multiply by 4 equals 12 which implodes to 3
6 multiply by 4 equals 24 which implodes to 6
And anything we multiply by by will always be 9.

x4_x7

In this ring set I have included black rings as spaces because the 3 6 and 9 rings are not created by the inner rings, However you can move around each ring clockwise multiplying by 4.

Multiplying by 5
When multiplying by 5 we create the doubling pattern in reverse.
1 5 7 8 4 2
This also creates the
3 6
oscillating pair.

Multiplying by 6
This creates the same crashing into 9 as did multiplying by 3.

Multiplying by 7
When we multiply by 7 we get the same family member groups as we did when multiplying by 4, but in reverse. We also get constant 3s and 6s.

Multiplying by 8
We create oscillating polar pairs when we multiply by 8
1 8
2 7
3 6
4 5

x8

When multiplying by a constant we create multiplying pairs similar to the polar pairs. The pairs are
1 is unique
2 and 5 (multiplying by 2 or 5 creates the same sequences just in reverse)
3 and 6
4 and 7 (multiplying by 4 or 7 creates the same sequences just in reverse)
8 is unique.
We will see this unusual pairings again.