Addition and subtraction with negative numbers

Leon has been on the beach all day. Now, he’s at home, emptying his pockets on the table. Some sand. A plaster. A tenner.

And a note. Minus fifteen - what does that mean? That’s right! Leon borrowed money from Lina to buy something to eat. So he owes Lina money.

He has a tenner. Plus that debt. And a debt, that decreases how much money he has. A debt is negative money. We write the negative amount in parentheses, to make it clearer.

Ten plus minus fifteen. It says plus minus, because we add a debt. We add negative fifteen. That’s the same thing as saying ten minus fifteen. Because, plus minus, that means just minus.

And ten minus fifteen, that is: Minus five, negative five! Even if Leon repays all the money he’s got, he will still owe Lina a fiver. Are you keeping up? Good. Let’s do some more calculations.

Here’s the value of all of Leon’s money and debt, taken together. Minus five. Leon has negative assets. He is in debt. Sad.

Uh oh, here comes Lina. And Leon doesn’t have enough money to pay back the money that he owes. But Lina is in a good mood, and says Leon doesn’t have to pay back what he borrowed. She says: “Let’s forget that debt, it’s on me!” Thank’s Lina, that was nice. But what does it mean for Leon’s money.

Now, we are removing a debt. We start with the value of Leon’s total assets and debts. Minus five. And then we remove something. To take stuff away we use subtraction, so let’s write a minus sign.

And the thing we are removing is a debt, which we write as minus fifteen. Oh. Minus five... ...minus minus fifteen. Does it seem weird? What can minus minus mean?

What does it mean to remove a debt? Pause the film and think. Talk to a friend if you can, to see if you think alike. Well, when Lina says, “You don’t owe me any money anymore,” then Leon is getting something from Lina. So it is plus-money.

And in mathematics we express it like this: Minus something negative means plus. Or a bit more formally: to subtract a negative number, is the same as adding the corresponding positive number. Let’s take another example, with more terms in it. 110 plus minus 50 minus 10 minus minus 40. Pause and calculate!

Remember the rule? ‘Plus minus’ means minus. And ‘minus minus’ means plus. So we have: 110 - 50 - 10 + 40 which equals ninety. Did you get the same result? If not, rewind the film and look again, to make sure you understand what happened.

To sum up: There are numbers lower than zero: negative numbers. Negative numbers are written with a minus sign in front, and within parenthesis. When you add a negative number, you see ‘plus minus’ in a calculation, it means the same thing as a just ‘minus’. When you subtract a negative number you see ‘minus minus’. That means ‘plus’.

And it’s this last rule that can be a challenge to master properly. Think of it as a debt that goes away. When you see ‘minus minus’, that’s someone saying “Don’t worry I’ll pay your debt!” And that’s something positive!