Adding and Subtracting Integers

Every adding and subtracting question is really a short journey along the number line. The first number tells you where to stand, and the second number tells you how to move. Look at negative seven plus three. You stand on negative seven, and adding three means three hops to the right, so you land on negative four. Now look at four plus negative six. You stand on four, but this time the journey is negative, so you hop six places to the left: three, two, one, zero, negative one, negative two. You land on negative two. Nothing special happens when you cross zero. It is only another tick you walk straight past.

Here is the first big rule, and a picture that makes it obvious. Five plus negative three means you start on five and add something worth negative three. Adding a negative amount pushes you the wrong way, so you hop three places left and land on two. But five minus three also means three hops left from five, and it also lands on two. The two questions are the same journey written two different ways, so adding a negative is exactly the same as subtracting. Think of money: being handed a three dollar bill leaves you three dollars worse off, just as if you had spent it. Tidy the two signs into one, then hop.

The second big rule looks strange until you see it with counters. Start with five positive counters, worth five. Now drop in two zero pairs, each one a positive and a negative sitting together. A pair like that is worth nothing, so your total is still five, but now there are two negatives on the table for you to take away. Take them, and seven positives are left. So five minus negative two is seven. Subtracting a negative is the same as adding. Money says it too: if the bank cancels a debt you owe, taking that negative away leaves you better off. So five minus negative two is five plus two, which is seven.

You now have both rules, so put them in one place. When two signs end up side by side, tidy them into a single sign before you do anything else. Two signs that are the same make a plus, and two signs that are different make a minus. That is why plus negative five becomes minus five, and minus negative five becomes plus five. From there the work never changes: start at the first number and hop. Take negative twelve minus negative five. The touching signs become a plus, so it is negative twelve plus five, which is five hops right, landing on negative seven. Long chains are fine as long as you tidy and hop one step at a time.

Word problems hand you the same two rules dressed up in a story, so translate before you calculate. Words like rises, warms, gains and pays in mean add. Words like falls, cools, loses and spends mean subtract. Questions asking how much warmer, how far apart or what is the difference always mean the higher number minus the lower one. How much warmer is five degrees than negative nine? That is five minus negative nine, which tidies to five plus nine, so fourteen degrees. Check it on the line: nine hops up to zero, then five more, which is fourteen hops in total. And if the balance is negative eighteen and you pay in twenty five, negative eighteen plus twenty five leaves you seven dollars.