Negative Numbers and the Number Line

Until now every number you have used has been zero or bigger. Look at the thermometer: the liquid has dropped below the zero mark, and the reading is five degrees below zero. You cannot write that as an ordinary counting number, so you write it as negative five. Negative numbers appear everywhere once you look. A car park level below the ground, money you owe the bank, the floor of a lake below sea level. In every case zero is just an agreed starting line, and the negative sign tells you which side of that line you are on. The whole numbers, their negatives and zero together are called integers, and they run on forever in both directions.

Here is the number line you already know, opened out to the left of zero. Zero sits in the middle. To its right the positive numbers count up one, two, three and onward. To its left the negative numbers count down in exactly the same steps: negative one, negative two, negative three. The gap between any two neighbours is always the same, so the negative side is a perfect mirror of the positive side. Negative three sits three steps from zero, and so does positive three, just in the opposite direction. When you see a number like negative six, read the sign as part of the number itself. It is one number, and it lives six steps to the left of zero.

Pick any number and walk the same distance to the other side of zero. Where you land is called its opposite. Two is two steps right of zero, so its opposite, negative two, is two steps left. Four pairs with negative four, six pairs with negative six, and the arcs on this line show each pair balancing across zero. Taking the opposite twice brings you straight back to where you started, so the opposite of negative seven is seven, and the opposite of that is negative seven again. Zero is the odd one out. It is nought steps from itself, so zero is its own opposite and counts as neither positive nor negative.

The number line turns adding and subtracting into walking. Adding moves you to the right, subtracting moves you to the left, and the number tells you how many steps. Start at three and subtract five. Hop left five times: three, two, one, zero, negative one, negative two. So three minus five is negative two. Nothing stops you at zero; it is just another tick you walk past. The same works from a negative start. From negative four, adding six means six hops right, which lands you on two. From negative two, subtracting three means three hops further left, landing on negative five. Once you can walk the line, you can always subtract a bigger number from a smaller one.

In real problems the number line is usually hiding behind the words, and your job is to find it. Ask two questions. Where is zero, and which direction counts as negative? For a building, zero is the ground floor and negative means underground. For altitude, zero is sea level and negative means below it. For money, zero is an empty account and negative means you owe. Once you have decided that, the sums are just hops. You have twenty dollars and spend thirty-five: twenty minus thirty-five is negative fifteen, so you owe fifteen. It is negative six degrees and warms by ten: negative six plus ten is four degrees. Same number line, four different jobs.