Adding and subtracting integers
Until now, numbers may have stopped at zero. Integers extend them in the other direction, into the negatives. Temperatures below freezing, money owed, floors below ground level, points lost in a game: negative numbers describe everyday situations all the time. This year you learn to compare, order and calculate with integers, and the number line is once again the clearest guide.
An integer is any whole number, positive, negative or zero. The negatives mirror the positives on the other side of zero, so just as 3 is three steps to the right, negative 3 is three steps to the left. Once you picture them on a line, adding and subtracting become simple movements rather than rules to memorise.
Moving along the number line
Adding and subtracting integers is a matter of direction. Adding a positive number moves you to the right, and subtracting moves you to the left. Starting at negative 2 and adding 5 means moving five steps right, passing through zero and arriving at 3. Starting at 2 and subtracting 6 means moving six steps left, dropping below zero to negative 4, exactly like a temperature falling below freezing.
This movement picture also makes ordering integers clear. A number further to the left is always smaller, so negative 5 is less than negative 1, which is less than 0. It can feel strange at first that negative 5 is smaller than negative 1, since 5 is larger than 1, but on the line negative 5 sits further left, and that settles it. Comparing integers is just reading their positions from left to right.
Subtracting a negative
The one rule that puzzles most students is subtracting a negative number. The result is the same as adding a positive: 5 minus negative 3 equals 5 plus 3, which is 8. The clearest way to feel why is to think of a negative as a debt. If you owe 3 dollars and that debt is taken away, you are 3 dollars better off, exactly as if someone had given you 3 dollars.
Adding a negative works the opposite way, moving you further left, so negative 3 plus negative 4 is negative 7. A simple way to keep track is to watch the signs: two minus signs next to each other turn into a plus, while adding a negative keeps you heading down. With the number line in mind and the debt picture to fall back on, integer arithmetic stops being a set of tricks and becomes a sensible story of moving left and right, which prepares you for working with negative numbers throughout algebra and beyond.