Multiplication with negative numbers

Leon and Lina sometimes borrow money from each other. For each loan they take, they note the amount on a piece of paper: a debt note, with a negative number, that corresponds to how much money the person holding the note owes the other person. Lina has several identical debt notes. Five of them, each with minus ten on it. The value of all five notes together is minus fifty.

Did you see what happened there? The minus sign was carried over to the result. When you multiply a positive number by a negative number, the product is always a negative number. Plus times minus equals minus. It doesn’t matter which order the factors come in a multiplication.

Plus times minus is minus and minus times plus is also minus. Lina has helped Leon to fix his bike all day, and Leon tells Lina she can throw away three of the debt notes, as a token of his gratitude. Being allowed to throw away debt. That’s like being given money! When we remove a debt, we remove a negative number.

And the negative number we remove is minus ten. And how many negative tens are we removing? Three of them, each worth minus ten. So removing three debts of ten each, can be stated as: Minus three times minus ten. Oh, now we’ll multiply two negative numbers.

What do you think? Will we get a positive or a negative number when we do this? Pause, and think for yourself. A debt is a negative value. And taking away debt is also negative.

We remove something negative. And removing something negative, is positive! Minus three times minus ten is thirty. Positive thirty. Because: minus times minus is plus.

The two minus signs cancel each other out. A negative number multiplied by another negative number always equals a positive number. Try these calculations for yourself. Pause after each calculation if you need to think, and see if you can get to the right answer, before you hear it: Minus two times minus four. Minus times minus is plus.

Two times four is eight. Positive eight. Four times minus five. Plus times minus is minus. Four times five is twenty.

Negative twenty. Minus eight times three. Minus times plus is minus. Eight times three equals twenty-four. Negative twenty-four.

Minus six times minus six. Minus times minus is plus. Six times six equals thirty-six. Positive thirty-six. Woah!

What’s this? Minus five times minus ten times minus two? Three negative factors! We haven’t covered that! Take it easy, and calculate the multiplications one at a time.

Minus five times minus ten is plus fifty. And what do you have left then? It reads plus times minus. A multiplication, with one positive and one negative factor. So, the product has to be negative.

Negative one hundred. Now try for yourself and make up your own calculation. Choose four numbers with a minus in front: four negative factors. And see if you can figure out whether the product is positive or negative. One multiplication at a time, and remember: minus times minus is always plus.