Multiplying and Dividing Integers

Multiplication is about scaling — 'lots of'. Two times three means two lots of three. Each arrow takes us three to the right — so we land on six. We've scaled. Doubled 3. When we multiply by a negative, we also scale by negative one — which flips the arrows. So negative two times three is still two lots of three, but the negative one flips it. We land on negative six. What about negative one times negative four? Start with one arrow of negative four — landing on negative four. The negative one flips it, taking us all the way to positive four.

So here are the rules. Positive times positive is positive. Positive times negative is negative, because the negative flips the direction. Negative times positive is negative. And negative times negative is positive, because flipping twice brings us back to the original direction. Notice the pattern: when the signs are the same, the answer is positive. When the signs are different, the answer is negative.

Negative three times four. Ignore the signs: three times four is twelve. Now the signs — we have a negative times a positive, which results in a negative. So negative three times four equals negative twelve.

Negative six times negative five. Ignoring the signs: six times five is thirty. Both signs are negative — same signs means a positive answer. Negative six times negative five equals positive thirty.

Five times negative eight. Five times eight is forty. We have a positive times a negative, which results in a negative. So five times negative eight equals negative forty.

Negative three squared. Squared means multiplied by itself — so this is negative three times negative three. Three times three is nine. Two negatives — same signs — so the answer is positive nine.

Negative five cubed. Cubed means multiplied by itself three times — negative five times negative five times negative five. Five times five times five is one hundred and twenty-five. Three negatives — the answer is negative one hundred and twenty-five.

Division follows exactly the same rules. Multiplication and division are two sides of the same coin. Same signs give a positive answer. Different signs give a negative answer.

Negative twenty-four divided by negative six. Ignore the signs: twenty-four divided by six is four. Same signs — the answer is positive. Negative twenty-four divided by negative six is four.

Twenty divided by negative four. Twenty divided by four is five. Different signs — so the answer is negative. Twenty divided by negative four is negative five.

Negative twenty-four over six. Remember — the fraction line means divide. Twenty-four divided by six is four. A negative divided by a positive gives a negative. So negative twenty-four over six equals negative four.