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Why is `(-1)(-1)=1` :

Here is one way to think about this:

Consider a real number line. Every point on the number line is associated with a number, called the coordinate. So there is a point to the right of 0 (the origin) that is associated with the number 2.

Instead of thinking of 2 as a point on the number line, you could think of it as a length -- the length of the segment from the origin to the point. If you stood at the origin and went to the coordinate 2, you would traverse 2 units. Also, and this is important, you would be facing to the right.

If you began at the origin and went to the coordinate -3, you would traverse 3 units, but you would end up facing towards the left -- towards negative infinity.

Now multiplication is a dilation -- it expands or contracts figures. So if you think of 6x2, you start with the "segment" from the origin to the coordinate 2, and stretch this segment to a length 6 times as long. Thus you end up at coordinate 12.

The thing to note is that when you are at 12, you are facing to the right.

If you multiply 6x(-2), you start at -2 (facing left) and stretch the sement and end up at -12, facing left.

Now what could multiplying by a negative number mean? In this analogy it means that you still stretch by the same magnitude, but you face the opposite direction. (Much of the confusion regarding negatives is that there are three interpretations, some of which overlap. The "-" sign can mean subtract (minus), negative, or opposite. Here we use the meaning opposite.)

So (-1)x(-1) can be thought of as: start with a segment ending at -1. Multiplying by 1 gives you a segment of length 1, while multiplying by negative one gives you a segment of the same length in the opposite direction, thus ending at 1 facing in the opposite direction, or facing right.

So 6x2 ends at 12 facing right

6x(-2) ends at -12 facing left

(-6)x2 is the same as (-1)x(6x2); 6x2 gets you to 12 facing right, then multiplying by -1 means you go to -12 facing left.

(-6)x(-2) is the same as (-1)(6x(-2)); 6x(-2) gets you to -12 facing left; then multiplying by -1 moves you to the opposite of -12 or 12 and facing the opposite direction. So you end up at 12 facing right.


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