Exploding Dots

11.4 Multiplication

Lesson materials located below the video overview.

This is where Napier’s brilliance starts to shine.


To perform multiplication, Napier suggested viewing the columns of the checkerboard as their own \(1 \leftarrow 2\) machines! This way, each dot in a box represents a product. For example, in this picture the dot has value the product \(16 \times 4 = 64\).

What is lovely here is that dots in the same diagonal have the same product value: \(64 \times 1 = 32 \times 2 = 16 \times 4 = \cdots = 1 \times 64\). So in addition to doing \(1 \leftarrow 2\) explosions horizontally and vertically, we can also slide dots diagonally and not change the total value represented by dots on the board.


Here’s a picture of one copy of \(19\) plus four copies of \(19\), that is, here is a picture of \(19 \times 5\).


Slide each dot diagonally downward to the bottom row: this does not change the total value of the dots in the picture. The answer \(95\) appears.

More complicated multiplication problems will likely require using a larger grid and performing some explosions. For example, here is a picture of \(51 \times 42\).


Sliding gives this picture

and the bottom row explodes to reveal the answer \(2142\).


Question:  One can do polynomial multiplication with the checkerboard too! (One needs two different colored counters: one for dots and one for antidots.) Do you see how this picture represents \(\left(x^{2}-2x+1\right)\left(x^{3}-2x+2\right)\)? Do you see how to get the answer \(x^{5}-2x^{4}-x^{3}+6x^{2}-6x+2\) from it?


Question:  How would you display the product \(\left(1-x\right) \left(1+x+x^2+x^3+x^4+\cdots\right)\)? What answer does it give?

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