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Station  11D

Napier did not introduce the notion of an antidot, but suggested performing subtraction this way instead.

To compute 10649106-49 , say, represent the larger number on the second row of the board and the smaller number on the bottom row.

I11S11D - Image01

Starting at the left of the second row, perform unexplosions so that each dot in the bottom row has at least one dot above it.

I11S11D - Image02

Now subtract dots from the second row, one for each dot that sits on the first row. We see the answer 5757 appear.

I11S11D - Image03

Question: The picture below shows how we performed subtraction in the 121 \leftarrow 2 machine using antidots. Can you see a correlation of the two approaches?

I11S11D - Image04

Question: Consider a 121 \leftarrow 2 machine fully loaded as shown.

I11S11D - Image05

Do you see if you perform all the explosions, all the dots disappear? This shows that, in some sense, the infinitely long base-two number 111112\cdots 111112 represents the number zero. (See the chapter on Some Unusual Mathematics for Unusual Numbers for more on this.)

This means we can add 11112\cdots 11112 to a picture of a negative number and not change the number. For example, we see that another representation of 49-49 in a 121 \leftarrow 2 machine is 11111001111\cdots 11111001111.

I11S11D - Image06

Thus every negative number can be presented in a 121 \leftarrow 2 machine without the use of any antidots. (The trade-off is that one must then use an infinite number of dots!)

Compute 10649106 - 49 in Napier’s checkerboard again but this time thinking of it as an addition problem, 106+(49)106 + \left(-49\right), that can be presented on the board using only dots.

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