Multiplying Negative by Negative
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 X-Sender: firstname.lastname@example.org 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) 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 From: Anonymous Date: Mon, 7 Nov 1994 14:40:57 -0500 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
From: Tom Rocklin Date: Tue, 7 May 1996 12:55:48 -0500 Subject: Multiplying negatives My son found the debt metaphor pretty helpful. I have since been told about one I like even better. Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps. So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
Date: 02/02/98 at 22:34:37 From: Jim Caprio Subject: Negative times negative My wife is going to love your site, as she is a high school math teacher (LaSalle Sr. High, Niagara Falls, NY), teaching algebra & geometry. Here's one that's taught in 7th grade, but is still impossible to explain effectively: if you know of an intuitive explanation for why the product of two negative numbers is positive, please share it with me (and I'll pass it on to my wife, because her SENIOR HIGH REGENTS STUDENTS are constantly getting confused about it!). I think their confusion is on what exactly does it "mean" to multiply something a negative number of times. Perhaps expanding the product out in the traditional manner might illustrate the "problem": 2 x 3 = 2 + 2 + 2 = 6 Multiply 2 three times - no problem 2 x (-3) = -3 + -3 = -6 Multiply -3 two times - no problem (-2) x (-3) = ? Confusion! How do you expand this? Thanks for the insight, Jim Caprio
Date: 02/03/98 at 01:26:27 From: Doctor Pete Subject: Re: Negative times negative Hi, Imagine the following situation: You're standing at a street corner. At exactly noon, a car passes by you, going east, at a constant rate of 20 miles/hour. After one hour has elapsed, that is, at 1:00 p.m., it would be 20 miles east of where you're standing. And one hour ago, at 11:00 a.m., it was 20 miles west of where you're standing. Now, what would it mean if the car was travelling east at -20 miles/hour? It'd be in reverse, going west, of course. So, if it were going at -20 miles/hour east, at 1:00 p.m. it would be 20 miles west of where you're standing. Finally, if it were going -20 miles/hour east, at 11:00 a.m., where would it be? 20 miles east, of course. What does this mean? (-20 miles/hour)(-1 hour) = 20 miles. Hence here is a "concrete" explanation of why two negatives multiply to make a positive number. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 09/11/2011 at 05:53:45 From: Mrs. Nehila Subject: Negative x negative Hi Dr. Math! I am a new teacher who will be teaching 7th and 8th graders multiplication with integers. I'm using your site to see what people commonly struggle with, and to generally verify my lessons, and so found Jim's question, above. Most of my kids seem to have it drilled into their heads that when two negative signs are together, they make a positive. So would it be correct to think of Jim's example this way? -(-3) + -(-3) = 6 In other words, by representing the -2 as an extra negative sign (which would be the opposite of +), you are now accounting for the (-2). I am going to try it out and hope it's as clear to my students as it seems to me!
Date: 09/11/2011 at 11:12:15 From: Doctor Ian Subject: Re: Negative x negative Hi Mrs. Nehila, I don't see how that really addresses the problem, since it's basically circular. That is, saying ... -(-a) = a BECAUSE -1*-a = a. ... doesn't really explain anything. Personally, I think of this topic as an opportunity to teach a much deeper 'intuition' about mathematics as a whole, rather than focusing on this one particular result. The intuition I have in mind is that the nature of math is this: We choose some definitions, and then we trace out the consequences of those definitions, to see where they take us. And the reason we do that -- what makes it fun, and interesting (at least for mathematicians, if not always for students in math classes) -- is that they sometimes take us to surprising conclusions! This is one of those times. Having defined a negative number as the additive inverse of a positive number ... a + -a = 0 ... we then see where that takes us. And one of the places it takes us is here: http://mathforum.org/library/drmath/view/55717.html That is, what is happening here is that our CHOICE of a definition for negative numbers FORCES a particular conclusion, which is that the product of two negatives is positive. And this is how math works. If you just think of it a bunch of rules that have been known forever, and that you have to memorize, you miss all the fun. To put this another way, the deepest intuition you can have about why the product of negatives is positive is that it's not something we DECIDED up front, but rather something that we are REQUIRED to accept because of the definitions we started with. If your students can really get a handle on that, it will make everything about math easier to understand, for as long as they continue to study math. I like to explain this kind of thing with an analogy to other games, like chess. It turns out that there are certain endgames in chess, where for example, if one guy has just a king and a bishop, and the other guy has just a king, the latter (apparently weaker!) player can play out a stalemate. (I don't know whether that's true; I don't really play chess. But what's important is that situations LIKE this exist. For now, we can just pretend that it's true.) Now, we could say that this is a 'rule of chess.' But it's not really like what we normally think of as 'the rules of chess' -- what pieces there are, how they're arranged at the start of a game, how they can move, and how you take turns. Those are STARTING rules. But the impossibility of forcing checkmate with a king and a bishop against just a king is a DERIVED rule. No one set out with the intention of making it impossible to force checkmate if you're down to a king and a bishop. It's just something that follows from the initial definitions -- and the first time anyone figured it out, and proved it to be true, I'm sure it was quite a surprise. In the same way, the rule that a negative number is an additive inverse is a STARTING rule. And the rule that the product of two negatives is positive is a DERIVED rule. And the first time anyone figured it out, and proved it to be true, I'm sure it was quite a surprise. :^D But if we blur the distinction between those kinds of rules, then students lose 'the element of surprise,' so to speak. Do you see what I mean? Having said all that, one major science application where this rule shows up is forces. For example, if I have two positive charges (these can be electrical or magnetic), or two negative charges, those forces repel; whereas a positive charge and a negative one attract. If I have particles with charges of 1 and -1, k * 1 * 1 F = --------- = positive force (repulsion) r^2 k * -1 * 1 F = ---------- = negative force (attraction) r^2 k * 1 * -1 F = ---------- = negative force (attraction) r^2 k * -1 * -1 F = ----------- = positive force (repulsion) r^2 To make this more intutive, you can let your students play with some bar magnets. They'll quickly notice, without your even telling them, that like poles repel, while unlike poles attract. That manipulative experience puts some real world context to the rule that the product of two negatives behaves the same way as the product of two positives. Would this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math
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