The carrom coin on a number line

A carrom coin is struck to move it to the right, each strike moving it a certain number of units depending on the force used. The coin starts at 0.

If it is struck twice, first moving 4 units and then 3 units, where does it end up? Clearly at 4 + 3 = 7 units from 0. In general, if the first strike moves it a units right and the second b units right, the final position is P = a + b.

Now let the coin be struck in EITHER direction, left or right. You could work through four separate cases — both right, both left, right then left, left then right — but that is clumsy.

A much better way is to model the line the coin moves along as a number line, and use positive and negative integers. Take rightward movement as positive and leftward movement as negative.

Suppose the first strike moves the coin 5 units right from 0 and the second moves it 7 units left. Then the first movement is 5 and the second is −7.

Add them exactly as before: 5 + (−7) = −2. So the coin ends at −2, which is 2 units to the LEFT of 0. The same formula P = a + b now covers all four cases at once, with no case-by-case work.

Notice what the integer is doing. By modelling the movements as numbers that may be positive or negative, we capture TWO pieces of information in one symbol: the distance, which is its magnitude, and the direction, which is its sign. When we say the movement is −4, the magnitude is 4 and the direction is leftward. Try it: if the first movement is −4 and the final position is 5, what was the second movement?