NegativesDate: 10 Jan 1995 12:12:27 -0500 From: Andy Chen Subject: Negatives I know this sounds like a dumb thing to ask, but can anyone explain negative sign multiplication? I know that two negatives make a positive, and one positive and one negative make a negative, but why? People have tried to explain this to me in the past, but I've never understood it completely. While you're at it, how about negative sign division too? Does the reasoning behind negative multiplication and division also account for the fact that subtracting a negative number equates to really adding it? Thanks for the help Andy Chen achen@walrus.mvhs.edu Date: 10 Jan 1995 13:56:26 -0500 From: Dr. Ken Subject: Re: Negatives Hello there! We're glad you're taking some time to go back and look at the foundations of mathematics. It just so happens that a very similar question to yours was asked before, so I'll forward that on to you. Date: Sun, 6 Nov 1994 13:09:25 -0500 From: Sally Spencer Subject: Multiplication of Positive and Negative Numbers Dear Dr. Math, I'm trying to make sense of these rules so that they'll be easier to memorize: Pos x Pos = Pos, makes sense. I've been doing it since 3rd grade. And I can even think of a situation. I get six birthday cards with $5 in each. Pos x Neg = Neg, I can think of a situation for this, too. I get four bills for $20 each so I'd owe money. But, Neg x Neg = Pos just doesn't make sense. Does it ever happen in real life? My teacher said that you could say it would be the opposite of Pos x Neg but that seems like cheating. It's not realistic. Thank you, Sally Date: Sun, 6 Nov 1994 13:32:02 +0900 From: Dr. Margaret Subject: Re: Multiplication of Positive and Negative Numbers Hi Sally! Thanks for writing to us. This is a difficult question. I wish that I had a good explanation of it. Someone else asked us this recently, so I am going to give you the response that Dr. Demetri wrote. The specific example was -6*-6. I am afraid this might seem like "cheating", too. So, one way to think about this is to take 6*(-6) (that is 6 times -6), find the result to this (which is exactly what you have above, ie -36) and then consider what -[6(-6)] is. This is nothing other than the negative of 6(-6) or, if you prefer, its opposite (opposite numbers are two numbers whose sum is 0- you may know this, but I said it just in case you don't know). Clearly, the opposite of -36 is +36, because -36 +36 = 0. Therefore, -6*(-6) = +36 I hope this helps. Please feel free to write back if you have any questions. -Margaret, Math Doctor on call From: Dr. Ken Date: Sun, 6 Nov 1994 13:32:21 -0500 (EST) Subject: Re: Multiplication of Positive and Negative Numbers Sally! Thanks for the question! It's great that you want to actually make sense of the situation instead of just doing it by rote. I'll try to give you an example for the Neg x Neg case, based on your receiving bills thing. Let's say you got five bills in the mail for seven dollars each. Then you're right, you'd have 5 x -7 more dollars, i.e. -35 more dollars, i.e. 35 fewer dollars. But what if you had _sent_out_ five bills instead of getting them? Then, in a sense, you've gotten negative five bills, so you have -5 x -7 = 35 more dollars than you used to have. Unfortunately, I can't think of another example right now to really drive the point home. But I'll keep thinking, and I'll try to get back to you soon! Also, one of the other Math Doctors here might jump in and reply if they think of anything clever. Thanks! -Ken Date: Mon, 7 Nov 1994 14:40:57 -0500 From: Dr. Steve Subject: Re: Multiplication of Positive and Negative Numbers Hi Sally, I think you've gotten some good answers and here's another variation, again using your own example with bills. Neg x Neg: Imagine that you buy five gift certificates worth $5 each and you pay for them using your credit card. As you point out below, you now owe money, so that's -$25. The bill comes from the credit card company, but I TAKE IT AWAY from you and insist on paying it. You now have $25 of gift certificates without having paid anything. Taking away a debt is analogous to negating a negative. Take away five debts of $5 (-5*-5) equals a gain of $25. -- steve Date: Mon, 7 Nov 1994 14:40:57 -0500 From: Dr. Ken Subject: Re: Multiplication of Positive and Negative Numbers About your second question, Andy, once we've defined multiplication the way we have, we're forced to accept the rules for division, too. Because we have Negative x Positive = Negative, we can divide both sides of the equation by the negative or the positive, and we'll get Negative = Negative/Positive and Positive = Negative/Negative. As far as a + (-b) = a - b, that's simply the definition of subtraction. See, we define addition first, then we define negative numbers, and then we define subtraction by saying "a - b = a + (-b)." And that's that. Here's an example: if I have five dollars and someone gives me a bill for three dollars (i.e., they give me negative three dollars), I'll have 5-3=2 dollars left, assuming I pay the bill. I hope this gives you a few more things to think about! -Ken "Dr." Math |
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