## Sunday, November 16, 2014

### Subtraction and Negative One

We introduce a new card.

This means remove a One, ie take away one black counter.

If the following occurs in the game then the next move is obvious

What if the following occurs? There are no counters to remove!!

Stick to the rule of adding to the board whatever is on the card.

We have just created a new number, shown using a white counter. Now there are two types of numbers black ones and white ones.

How to name the new number!

This image is the negative of

Call the new number Negative One.

Sometimes because it is the result of the action of taking away One it is called Minus One.

In Maths we write (-1) for this number.

We have a new number and a new card .

The card can played in two ways:
1. Remove a One from the board - think of the card as showing a hole where the black counter has been removed;
2. Add a to the board.
NEW RULE- However you play a card the result is the same.

The following leads to two outcomes

has no counters on it, it is empty, nothing on it at all – it shows the number Zero

also shows the number Zero.

It may seem surprising that something , a black and white pair, is used for Zero but something is always used for nothing, the word nothing is not nothing, it is a word
You will already be familiar with using 0 for zero so why not ?

 Recording the Game List                 1                -1                 0 Maths 1 - 1 = 0 List               1            (-1)               0 Maths 1 + (-1) = 0

NEW RULE – at any time pairs of black and white counters can be removed or added to the board without changing the number on the board.

We have that is the negative of and so is the negative of  .

As well as meaning add a black counter the card can be thought of as meaning remove a white counter – think of the card as showing a black hole where a white counter has been removed.
If the game position is as below there are also two ways to play this.

The obvious way, by adding the black counter

The alternative is to remove a white counter. To have a white counter to remove first put Zero on the board.

 Recording the Game List            -(-1)                1 Maths  -(-1) = 1

The negative of Negative One or Negative Negative One, is One.

-(-1) = 1

In playing the Game it is more usual to add negative counters rather than removing counters.

Every number has its negative.

The negative of is and the negative of is .

We can see that Negative Four is four Negative Ones by considering this play.

 Recording the Game List           4       -(-4)           0 Maths 4 + (-4) =0