Why a number and its opposite always add to zero — and why −3 + 3 can never be 6.
Your phone is at 40 percent. You plug it in and it climbs to 60. Then you forget the charger, use it all afternoon, and it drops back to 40. How much did the battery change, start to finish?
Nothing. Up twenty, down twenty, back where you started. Plenty happened in between — but the net change is zero.
That is the whole idea of these notes. Every situation on these pages has the same shape: something goes one way by a certain amount, then the other way by the same amount, and the two cancel out completely.
Two numbers are opposites when they sit the same distance from zero on the number line, but on opposite sides. The opposite of 4 is −4. The opposite of −12 is 12. The opposite of 0 is 0, because it is already in the middle.
The sign in front of a number is not decoration. It is a direction. A positive number points one way from zero; a negative number points the other way. Four and negative four are the same size — the same four steps — aimed in opposite directions.
Stand on zero. Take three steps to the right. Now take three steps to the left. Where are you?
Back on zero. Not because nothing happened — you walked six steps — but because the two walks undid each other exactly. In symbols:
A number and its opposite together are called a zero pair, because zero is what they make. It works for every number there is. Seven and negative seven. One-half and negative one-half. Four hundred and negative four hundred.
Some people would rather see this with counters than with a number line. A yellow chip is worth +1. A red chip is worth −1. Put one yellow and one red together and they cancel — that pair is worth zero, so you can sweep it off the table.
Lay out three yellow chips and three red ones. Pair each yellow with a red. Every pair is worth zero. Sweep all three pairs away. What is left on the table? Nothing. Zero. The table is telling you the answer to −3 + 3, and there is no arithmetic to get wrong.
Both pictures — the walk on the number line and the chips on the table — are saying the same thing. Use whichever one makes sense to you. Most people end up with a favorite.
Once you know the shape to look for, zero pairs are everywhere. Each of these is a number plus its opposite. In every one, something rises by an amount and then falls by the same amount — or goes out and comes back in.
Notice that in every one of these, a lot happened. The hiker got tired. The money moved. The ball got carried. But the net change — the difference between where you ended and where you started — is zero.
A submarine dives to 120 meters below the surface, then rises 120 meters. Describe its final position using a zero pair.
Going down is negative and coming up is positive. The dive is −120 and the rise is +120:
The submarine is back at the surface. Its depth is 0 meters. It traveled 240 meters in total, and none of that changes the answer, because the question asked where it ended up, not how far it went.
Here is what goes wrong more than anything else. A student sees −3 + 3, notices two threes, and adds them: 6. Or applies a half-remembered rule about negatives and writes −6. Both answers ignore what the minus sign is for.
The minus sign is a direction. −3 means three steps to the left. Adding 3 means three steps to the right. There is no way to take three steps left and three steps right and end up six steps from where you started. The only place you can be is where you began.
The check: if you add a number to its opposite and get anything other than zero, you added the sizes of the steps instead of following the steps. Go back to the number line and walk it.
What if the two numbers are not opposites? Say −5 + 3. The chips still work. Lay out five reds and three yellows. Pair up what you can — three pairs cancel to zero — and look at what is left over.
That is the next skill. You do not have to master it yet — just notice that it is built entirely out of this one. Every sum with signed numbers is a matter of sweeping away the zero pairs and reading what is left.
Zero pairs are small, but they are load-bearing. In a few weeks you will subtract negative numbers by adding the opposite — and that only works because a number and its opposite cancel. Later in the year you will solve equations like x + 5 = 12 by adding −5 to both sides, and the 5 disappears for exactly the reason on this page. Get this one solid and those will feel obvious instead of magic.
You are ready to move on when you can answer these out loud, in your own words, without looking back: