Turning from addition and subtraction, how do you do multiplication and division with negatives? Actually, we've already covered the hard part: you already know the "sign" rules:
plus times plus is plus
(adding many hot cubes raises the temperature)
minus times plus is minus
(removing many hot cubes reduces the temperature)
plus times minus is minus
(adding many cold cubes reduces the temperature)
minus times minus is plus
(removing many cold cubes raises the temperature)
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The sign rules work the same way for division; just replace "times" with "divided by". Here is an example of the rules in division:
(Remember that fractions are just another form of division! "Fractions are division"!)
Some people like to think of negative numbers in terms of debts. So, for instance, if you owe $10 to six people, your total debt would be 6 × $10 = $60. In this context, getting a negative answer makes sense. But in what context could dividing a negative by a negative (and getting a positive) make any sense?
Think about having a snack at a café. When you go to pay, the kid has trouble running your debit card. He swipes it six times before finally returning the card to you. When you get home, you check your bank account online. You can tell from the amount that, yes, he actually charged you way more than once. Some portion of that total debit (being a negative on your account) is wrong.
You want to confirm the number of over-charges before you call your bank to correct the situation. How can you figure this out? You can divide the entire amount (let's say, $76.02) by the amount shown on your receipt (say, $12.67), which is the amount of one charge. Each charge is a minus on your account, so the math is:
(- $76.02) ÷ (- $12.67) = 6
So there were indeed six charges in total. The number of charges, 6, being the counting-up of the number of events, should be positive. In this real-world context, dividing a minus by a minus, and getting a plus, makes perfect sense. And now you know to direct customer service to cancel exactly five of the charges.
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You may notice people "canceling off" minus signs. They are taking advantage of the fact that "minus times minus is plus". For instance, suppose you have (–2)(–3)(–4). Any two negatives, when multiplied together, become one positive. So pick any two of the multiplied (or divided) negatives, and "cancel" their signs:
I'll start by cancelling off one pair of "minus" signs. Then I'll multiply as usual.
(–2)(–3)(–4)
= (–2)(–3)(–4)
= (+6)(–4)
= –24
If you're given a long multiplication with negatives, just cancel off "minus" signs in pairs:
The first thing I'll do is count up the "minus" signs. One, two, three, four, five, six, seven. So there are three pairs that I can cancel off, with one left over. As a result, my final answer should be negative. If I come up with a positive result, I'll know I've done something wrong.
(–1)(–2)(–1)(–3)(–4)(–2)(–1)
= (–1)(–2)(–1)(–3)(–4)(–2)(–1)
= (+1)(+2)(–1)(–3)(–4)(–2)(–1)
= (1)(2)(–1)(–3)(–4)(–2)(–1)
= (1)(2)(+1)(+3)(–4)(–2)(–1)
= (1)(2)(1)(3)(–4)(–2)(–1)
= (1)(2)(1)(3)(+4)(+2)(–1)
= (1)(2)(1)(3)(4)(2)(–1)
= (2)(3)(4)(2)(–1)
= 48(–1)
= –48
I got a negative answer, so I know my sign is correct.
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Here's another example, showing the same cancellation process in a division context:
The major difficulty that people have with negatives is in dealing with parentheses; particularly, in taking a negative through parentheses. The usual situation is something like this:
–3(x + 4)
If you had "3(x + 4)", you would know to "distribute" the 3 "over" the parentheses:
3(x + 4) = 3(x) + 3(4) = 3x + 12
The same rules apply when you're dealing with negatives. If you have trouble keeping track, use little arrows:
← swipe to view full image →
I need to take the 3 through the parentheses:
3(x – 5) = 3(x) + 3(–5) = 3x – 15
Here, I'll be taking a "minus" through the parentheses; I'll be distributing the –2 onto the x and the minus of the 3.
–2(x – 3) = –2(x) – 2(–3) = –2x + 2(+3) = –2x + 6
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Notice how I was careful to keep track of the signs inside the parentheses. The "minus" was kept with the 3 by use of another set of parentheses. Do not be shy about using grouping symbols to keep your intended meaning clear, both to the grader and to yourself.
The other trouble, related to the previous one, is with subtracting a parentheses. You can keep track of the subtraction sign by converting the subtraction to a multiplication by negative one:
I'll start by writing a little "1" in front of the parentheses. Then I'll draw arrows from this 1 to the terms inside the parentheses, to remind myself of what I'm needing to do.
← swipe to view full image →
Don't be afraid to write in that little "1" and draw in those little arrows. You should do whatever you need to do to keep your work straight and consistently get the right answer.
I'll work from the inside out, simplifying first inside the inner grouping symbols, according to the Order of Operations. So the first thing I'll do is take the –4 through the brackets. Then I'll simplify; I'll continue by putting a 1 in front of the parentheses and, to help me keep track of that -1 that I'll be distributing, I'll draw my little arrows.
← swipe to view full image →
This one is tricky. They're having me subtract a fraction. I need to combine the fractions, which means combining the numerators. To make sure that I don't lose track of exactly what that "minus" means (namely, that I'm minus-ing the whole numerator of the second fraction, not just the x), I'll convert the minus to a plus of a –1:
← swipe to view full image →
Note that I converted from subtracting a fraction to adding a negative one times a fraction. It is very easy to "lose" the minus when you're adding messy polynomial fractions like this. The most common mistake is to put the minus on the x and forget to take it through to the –2. Take particular care with fractions!
For extra practice with parentheses, try here.
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