Division with negative numbers

It’s the first day of summer holidays, and Leon, Lina, and Michael have just had ice creams at a café. It’s time to pay and they get the bill. Wow, those were expensive ice creams! Yup, this is how much they owe the café. A debt, that is.

We can express a debt using a negative number, like this. Now they are to split the bill between them. Here's how: Take the total amount. Divide it by the number of friends, three, and you get how big a debt each one of you has. That seems correct.

The quotient ought to be a negative number, because each of the friends has a debt. A negative number, divided by a positive number, is always a negative number. Minus, divided by plus, equals minus. Leon thinks this seems strange. Can he possibly have eaten that much ice cream?

Well, look here Leon, we can calculate the other way too. Take the shared debt. And divide it by the amount that each of you owes. Your combined debt, divided by how much each of you individually owes, gives how many there are of you. Positive three.

Because there are three of you, not minus three. Three. Does this resemble anything you have done before? Maybe you see a similarity here, a parallel, with how to multiply negative numbers? That’s no coincidence, because multiplication and division are closely related.

Division is multiplication backwards. Follow this here, we’ll take it slowly, step by step: Minus 20 divided by 4 equals minus 5. Here, it’s easy to see which number the minus sign belongs to, so here we don’t need parentheses. Let’s do one more. Sixteen divided by minus two equals negative eight.

Minus twenty four divided by minus four equals six. A positive six. Can you see which rules we have followed here? ‘Minus divided by plus’ is minus. ‘Plus divided by minus’ is also minus. But ‘minus divided by minus’ is plus. Do you recognise the pattern from the other operations?

We can state it like this: Two same signs are positive. This is valid for addition and subtraction, for multiplication, and for division. Two different signs are negative. And that’s valid for all operations and regardless of the order of the signs. But it’s one thing being able to calculate, and another thing being able to pay.