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Beg Algebra: Multiplying Real Numbers

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  • Type: Video Tutorial
  • Length: 7:50
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 84 MB
  • Posted: 06/27/2009

This lesson is part of the following series:

Beg Algebra: The Real Number System (11 lessons, $14.85)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/beginningalgebra. The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Multiplying Real Numbers
Well, now I want us to review something that we've heard about a long time ago, and I really want us to try to bring some understanding to, and that is how we multiply numbers with various sign types. Now, you all, in fact, might remember--I sort of vaguely remember this from my youth, my mathematical youth--that if we're multiplying two numbers that have the same sign, then the answer is always a positive. If we multiply two numbers that have different signs, then the answer is always a negative. Now, here, I want us to think about this together and to come to some reason to see that in fact some of these things make sense.
Now each of these things actually has two principles, and let's just make sure we're clear on this. Multiplying same signs together; that means that if we take a positive number and multiply it by a positive number, the resulting product will be positive. Well, that makes complete sense. We can see that, actually, if we think about things in terms of counting. If we take three things and multiply it by two, then we have six. We have 3 and 3 more, so we have a positive number of them. So that makes complete sense, and that is going to be our foothold for understanding these other ones.
The second one that comes under this rubric is that if we take a negative number and multiply it by a negative number, then that resulting product is still positive. And this, by far, is the strangest one, and you know what, if you are like me, you might not have ever seen a reason why a negative times a negative is a positive. In the next lecture, I'm actually going to show you really fast the basic underlying idea, and it's really sort of interesting, but for now, we're going to hold off on that explanation.
The second sort of template is the product of different signs. So what that means is if you take a positive number and multiply it by a negative number, the resulting product is going to be negative. Similarly, if you take a negative number and multiply it by a positive number, again, the resulting product is negative.
So let's take a look at three of these in particular, and save the negative times the negative explanation for the next little short lecture that we'll see together. So let's just look at some examples, and of course, they start off really easily if you just start with two numbers. Like, let's say (3) o (5). Well, they're both positive, and so, therefore, we get a positive answer. Not a big deal. Now by the, by this principle, we can actually see why a negative multiplied by a positive would, in fact, yield a negative answer. So let's take a look at that together. Suppose that I take (-3) o (5). Well, we know the answer if we remember the rule; a negative times a positive is a negative, so this is just using the rule, which I'm not a big fan of. I want both you and I to really understand what is going on here. So now let's see why that makes sense. What is -3? Really, we could think of -3 as just -1 in front of the 3, and then if I put back the 5, then I can use the property that if you want to multiply a bunch of numbers, you can multiply them in any order you want. So I'm going to multiply these two first. If you multiply these two first, then you see 15. So what I see here is -1 in front of a 15, but what's a -1 in front of a 15? Well, that's just -15. That's a way of thinking about it where it makes a little bit of sense why a negative times a positive is a negative.
Another way of thinking about it, by the way, which captures another great theme of ours, is the following: combining like terms. What if you take this quantity, we're multiplying it by 5. That means we add it to itself 5 times. That means--see, here's a different way of thinking about it--this is -3 + -3 + -3 + -3 + -3, and how do we perform this? You can think of this visually if you want. I start at 0 and then I move over. In fact, I even have a little number line here. It won't go all the way, but it'll take us part of the way. Start at 0, -3 means we go to 3 and then we go -3 again, and that gets us to -6, and -3 again gets us to -9, and -3 again gets us to -12, and another -3 gets us to -15. So you can see all correct roads lead us to -15. Again, here we're combining like terms. So that's the idea of why a negative times a positive is always a negative.
All right. Now let's take a look at an example that is a lot stranger. How about (-2)(-7). Well, since I have a negative and I'm multiplying by a negative, the resulting product will be positive, and it will be 2 o 7 which is 14, and this is so odd, and it really leads to the question why, and that's the point of the next little short lecture--we'll see exactly why. But if we take this as a fact, for the moment, then we actually get sort of a fun little consequence, which is that if we ever have a bunch of negative numbers that we're combining, we can actually say something about the sign.
See, for example, suppose that I take a negative number, I'll call it (-a) and another negative number, I'll call it (-b), and a (-c), let's take a look at the product of those three numbers. Well, what do we know? Well, a negative times a negative will get me a positive. So, in fact, this product here, this first product, will give me a +(ab) and I multiply that by the (-c) and what do I know? I know that a positive times a negative is a negative. So I get (-abc), and this principle actually illustrates an interesting phenomena; that if we have an odd number of factors that we're taking the product of and each one is negative, well then the first two, negative times a negative makes a positive. The next two, negatives times a negative makes a positive, but since there is an odd number, there will be one last factor without a mate. So all the other ones will be positive--positive times one negative yields a negative. So what's the moral? If you have an odd number of negative numbers that you're multiplying, then, in fact, that resulting product will always be negative.
Similarly, if you have an even number of negative numbers, then they pair up perfectly like Noah's Ark, and, in fact, each pair is positive, and so, therefore, the resulting product there will be positive.
So, there you can see sort of how the signs fit into the mix in terms of multiplying numbers, and in the next presentation we will finally answer the question why is it that a negative times a negative is a positive. I'll see you there.
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