Multiplying and Dividing Integers

You already know that multiplying is a fast way of adding the same thing over and over. Three times five is five plus five plus five. Nothing changes when the thing you are adding is negative. Four times negative three means four lots of negative three, so start at zero and jump three to the left, four times. You land on negative twelve. Adding negatives always drags you left, so a positive number of negative lots always gives a negative answer. Order never matters in multiplication, which is handy, because you cannot picture negative three lots of anything. If you meet negative three times four, just swap it round and picture four lots of negative three instead.

Rather than jumping along a line every time, use the sign rules. When the two signs are the same, the answer is positive: positive times positive is positive, and negative times negative is also positive. When the two signs are different, the answer is negative: positive times negative, or negative times positive, both give a negative. That is the whole rule. The safest way to use it is to treat every question as two separate jobs. First multiply the digits and ignore the signs completely, so seven times eight is fifty-six. Then look back at the signs and decide. Seven and negative eight have different signs, so the answer is negative fifty-six.

Negative times negative being positive can feel like a trick, so here is why it has to be true. Start with three times negative four, which is negative twelve. Drop the left number by one each time. Two times negative four is negative eight, one times negative four is negative four, and zero times negative four is zero. Look at the answers going down: negative twelve, negative eight, negative four, zero. Each one is four more than the one above. The pattern has no reason to break, so the next line, negative one times negative four, must be four, and the line after that must be eight. That is exactly what the rule says. Negative times negative is positive.

You do not need a second set of rules for dividing, because dividing simply undoes multiplying. Since negative four times five is negative twenty, it must be true that negative twenty divided by five is negative four, and that negative twenty divided by negative four is five. Read that fact family and the pattern falls out. When the two signs are the same the answer is positive, and when they are different the answer is negative — exactly as for multiplying. So work the same way. Divide the digits first and ignore the signs. Thirty-six divided by nine is four. Then check the signs: negative thirty-six and positive nine are different, so the answer is negative four.

When a question strings several numbers together, you do not have to track the sign at every step. Just count how many negative numbers there are. An even number of negatives gives a positive answer, and an odd number gives a negative one. Negative two times three times negative five has two negatives, so the answer is positive thirty. Add another negative and you get negative thirty. Work the digits out separately and then stamp the sign on at the end. These rules earn their keep in real situations. A diver dropping three metres a minute for eight minutes is eight lots of negative three, which is negative twenty-four. A thirty-six dollar debt shared between four people is negative thirty-six divided by four, so nine dollars each.