Negative Numbers on a Number Line

A number line does not stop at zero. It carries on to the left, and the numbers there carry a minus sign: negative one, negative two, negative three. You already meet them: a freezer at negative eighteen degrees, a diver twelve metres below the surface, a bank account that owes thirty dollars. Negative seven sits seven steps to the left of zero, exactly as far from zero as seven is on the right. The minus sign is part of the number's name, not an instruction to subtract anything.

On the line, smaller is to the left. That is true on both sides of zero, and it is the thing that trips people up: negative nine is further left than negative two, so negative nine is smaller. Nine is bigger than two, but negative nine is colder than negative two. Every negative number is smaller than zero, and smaller than every positive number. So to order temperatures from coldest, walk along the line from the left: negative nine, then negative two, then zero, then four.

A temperature that rises moves right along the line; one that falls moves left. From negative four, rising nine degrees: the first four steps bring you to zero, and there are five steps left, so you land on five. From six, falling ten: six steps to zero, four more steps to negative four. Whenever a move crosses zero, split it there — spend what it takes to reach zero, then carry on with what is left. Chains of moves work the same way, one move at a time: a lift at level negative three goes up five to two, then down seven to negative five.

The difference between two numbers is the distance between them on the line. When they sit on opposite sides of zero, that distance is the two parts added together: from negative six to zero is six, from zero to five is five, so the difference is eleven. When they are on the same side, subtract: from negative nine to negative two is seven. The classic mistake is to write five minus six, get negative one, and call that the difference. Negative one is not eleven. When in doubt, mark both numbers and count the gaps.

Two harder questions use the same picture. Halfway between negative twelve and eighteen: the distance is thirty, half of that is fifteen, and fifteen steps from negative twelve lands on three. And working backwards: a temperature rose fourteen, fell nine and finished at negative two — where did it start? Undo the moves in reverse order. Undo the fall of nine by rising nine: negative two becomes seven. Undo the rise of fourteen by falling fourteen: seven becomes negative seven. Every rise is undone by a fall of the same size, and you always undo the last move first. Quiz and exercises now.