## 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?

## Books

Take your understanding to the next level with easy to understand books by James Tanton.

BROWSE BOOKS

## Guides & Solutions

Dive deeper into key topics through detailed, easy to follow guides and solution sets.

BROWSE GUIDES

## Donations

Consider supporting G'Day Math! with a donation, of any amount.

Your support is so much appreciated and enables the continued creation of great course content. Thanks!