Big Picture Overview
This is the quick guide to the video. For more complete details watch video 8.
To give a big picture overview of arithmetic
To thereby show how long division fits in as repeated subtraction
The Big Picture Overview of Arithmetic
Circlemaths tends to view number as emerging from a starting point and evolving or growing
into the various different categories much as a seed grows to become a tree, in a naturally unfolding organic and holistic manner.
Counting is the start of arithmetic. And it breaks into counting up and counting down.
Henceforth all of arithmetic must follow the up/down division.
Counting is uniform.
It increases (or decreases) by a count of 1 exactly each time.
0 1 2 3...
...3 2 1 0
Counting (up) goes by one's
Adding And Subtracting
Addition is not uniform.
We can add by large and varying amounts.
In the example above counting is in black, adding in red.
We can add 2 + 4 = 6.
Likewise we could add 111 + 222 = 333
without having to count one by one
to check that we got there.
Counting up turns into addition and
counting down turns into subtraction.
They do this by smoothly blending and turning into each other.
Consider this series and ask is it adding or counting?
0 + 1 = 1
1 + 1 = 2
2 + 1 = 3
In one way it is adding, because it is adding on 1 each time.
At the same time it is counting.
Because counting is adding on one's.
The two turn into one another at this point.
Subtracting is the exact opposite counterpart to addition
and grows out of counting down.
that subtraction undoes addition,
just as counting down undoes counting up:
5 + 3 = 8
8 - 3 = 5
In like manner multiplication grows out of addition.
Here are some addition sums:
3 + 1 + 5 = 9
2 + 2 + 2 = 6
4 + 5 + 1 = 10
The middle one is special.
It adds the 2 to itself repeatedly.
It is repeat addition.
We can shorthand it by saying:
three two's make 6
3 x 2 = 6
We can draw that:
so you can see the three 2's.
Multiplication is just repeat addition.
Likewise division grows out of repeat subtraction.
3 x 2 = 6
6 / 2= 3
and we can draw that:
The black shows three 2's coming to 6
The red shows that there are three 2's which can be taken
off 6 before finally reaching zero.
6 / 2 = 3
Just as counting down undoes counting up
and subtracting undoes addition
division undoes a prior multiplication.
Division is no more no less than repeated subtraction.
The "Add a Complement" strategy works
because it applies first principles.