Exploding Dots

1.5 Subtraction

Lesson materials located below the video overview.

Here’s another story that isn’t true.

When I was a young child playing in a sandbox I discovered the arithmetic of the positive counting numbers through piles of sand. For example, I discovered that “two plus three” equals five with piles in a sand box:


And hours of fun was to be had with me counting piles, and combining groups of piles to explore addition.

But then one day I had a truly astounding epiphany! Instead of making just piles of sand, I realized I could also make holes. And I realized too that a hole is the opposite of a pile: Place a pile and a hole together and they cancel each other out.


When I was in school I was taught to call a hole \( -1\), and two holes \(-2\), and so on (but notation something like \(opp 1\) and \(opp 2\) would be much better!) and was told to do this thing called “subtraction.”

I’ve never believed in subtraction. I realized with my piles and holes that:

            Subtraction is just the addition of the opposite. 

I would read \(5 – 2\)  as “\( 5 + -2\)”, five piles PLUS two holes, to which the answer is clearly three piles.


And problems like \(7 – 10\)  presented my with no issue in elementary school, seven piles and ten holes clearly leaves three holes: \(7 +  -10 = -3\).


Comment: To read more about my “Piles and Holes” approach to arithmetic (including the multiplication of negative numbers!) see Chapter 4 of THINKING MATHEMATICS! Vol 1: Arithmetic = Gateway to All available at www.lulu.com. Also see lesson 1.4 of the ASTOUNDING POWER OF AREA course on this site.


SUBTRACTION IN \(1 \leftarrow 10\) MACHINES

In this course we have been working with dots, not piles. So … What’s the opposite of a dot?

Answer: Not sure.

But whatever it is, let’s call it an “anti-dot” and note, that like a pile and hole (or matter and antimatter, for that matter!), whenever a dot and anti-dot come together they annihilate and leave nothing behind. We’ll draw dots as solid dots and anti-dots as hollow circles:


So like piles and holes we can conduct basic arithmetic with dots and anti-dots:


Now let’s get serious. Consider, for example, the problem:


This is an addition of dots and anti-dots problem:



Oh. Easy!


As another example we see that \(423 – 254\), represented by diagram: 


has answer:


That is:


This is absolutely valid mathematically, even though the rest of the world may have difficulty understanding what “two hundred and negative thirty negative one” means! To translate this into familiar terms we can “unexplode” one of the solid dots to add ten solid dots to the middle box.


This gives the representation: \(1 | 7 | -1\). Unexploding again gives: \(1|6|9\). (Why?)


Thus we have:

\(423 – 254 = 169\).


Question 25: How were you taught to answer a problem like the following in school?ex1513

What is actually happening when you “borrow” a digit from another column?



Question 26: Compute each of the following two ways. First, the traditional school way by starting at the right and borrowing digits. Then, do it by starting at the left and moving to the right, using negative numbers in the answer.ex1514





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