20160723 CALCULATION, ACCUMULATION, SUMMATION:

Are they addition?

The mystery "add together" operation is known to Wikipedians as summation. You may even see this link on the "Multiplication" page. Well, multiplication is summation of the same number. Why don't they explain it this way?

Summation, according to Wikipedia, to me is no less weird than multiplication. Looks like the writers were struggling with another instance of the "stick and stake" dilemma.

Addition is a binary operation and it involves two stakes. The number of additions is always 1 less than the number of the addends. Wikipedians are trying to avoid instructions like: skip 1, add any two numbers and count 2, add a new number to their sum and count 3... messy, indeed.

There is an obvious solution: begin adding from 0 and count from 1. In

0 + 4 + 4 + 4 

the number of the + signs is equal to the number of the 4s. Why don't they teach this way?

Please start Windows Calculator (I believe, any other calculator will do) and type +4. It becomes 0+4. Now press "=" three times and get 12.

Windows Calculator and the similar devices (including the earliest mechanical calculators and my Original-Odhner) are accumulator machines. You got to clean (or zero) the accumulator before starting using it, so chains of additions start from 0.

In positional notation we omit the leading and the trailing zeros, but we remember they are here. For example,

   1
+100

we understand as

 001
+100

Similarly,

   7
*100

is actually

   7.00
*100.

and we move the point to make 700.

Can we understand

+N

as

0 +N

and

-N

as

0 -N

?

If we could, we could consider addition of three 4s as increment by 4 three times. It might look like

+4 +4 +4 

and sound like add four, add four, add four.

By the way,

0 -N

is negation, or just -N.

Is it strange to add without knowing the augend? To understand how it is possible, consider a simple positional shnumber machine presented in STEREOLEARNING. Click the FORWARD control in the bottom right corner to follow the explanation. A mechanical calculator is simply more complex (and much simpler to use).

By the way, like with any simple calculator, in SHNUMBERS we clean up the board and explicitly add the first quantity to 0. If empty spots and sheets of paper mean 0, we may think we add the first number by writing it. Are they teaching this at schools?

The odometer described by the Roman writer Vitruvius was supposed to count the distances dropping the pebbles in the accumulator bucket. One may imagine dropping several pebbles at a time. To accumulate those pebbles (or calculi), there was absolutely no need to know how many pebbles already were in the bucket. This was true addition.

I wonder why the engineers of Antiquity used the pebbles at all. Why didn't they draw the numbers on the wheel and take the readings? Was it easier to bring the bucket to the masons and order "this many" milestones?

The shnumber devices like abaci and calculators do use the augends. In a sense, they discover them while adding. The reason is: this devices receive the inputs and return results using numerals and/or positional notations.

Teaching unorthodox notations like

+4 +4 +4 

is not a wise policy for a parent, but I have a story to tell. 3 years ago I was not sure how to teach TLG's big sister long multiplication. I didn't see much problems with teaching it, I just wanted to train her in a method endorsed by the educators in her country. It seemed that educators in the US were not endorsing any method and abandoned any hope to teach multiplication at all.

Eventually, I used a periodic discount to buy the video course "Mastering The Fundamentals of Mathematics" by professor James A. Sellers. I did not know US universities teach elementary math. It turned out, they do, and they do know multiplication.

My biggest concern was how to add the partial multiplications. I did not remember what they taught us at school, but I suspected in the US they were doing it differently. The writing media in my school were dip pens (complete with portable inkwells, believe it or not). US schools were relying on pencils and erasers.

I learned a lot from that video course. First of all, professor Seller told me that fifteen times three is fifteen three times. Then, to my astonishment, I realized that he was using pencil and eraser like any elementary school teacher. He was writing the carries in the same place, on top of the numbers. I could not believe they didn't let the students to keep track of their mistakes.

The biggest of his multiplications was 3x3 (456x123). When it came to adding 3 "long" numbers, I was holding my breath. OK, said professor, I taught you guys this stuff in my first lecture about addition.

He only taught addition of 2 long numbers. I had to assume that adding 3 long numbers at once was not in his curriculum at all.

Last Autumn, when I asked TLG to do long addition, she said she wanted to add 3 numbers. I muttered that three numbers are two numbers plus one, and only then I clicked. Adding "together" five or six long numbers, position by position, in accumulator mode: Yes, I did practice this. It was 51 or 52 years ago. So much progress in such a short time.

I picked two 10-digit numbers from TLG's stock and multiplied them by hand. My calculation is in the beginning of this page. I found only one acceptable way to keep track of the carries, and I suspect I would have done better with a dip pen. I simply did not have it. I used a ballpoint pen.