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Int Algebra: Properties, Identities, & Inverses

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  • Type: Video Tutorial
  • Length: 11:34
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 124 MB
  • Posted: 06/26/2009

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, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/intermediatealgebra. The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.

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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In algebra what we're constantly going to be doing is trying to manipulate numbers in order to solve things. We'll be given questions. We'll be given conundrums. We'll be given puzzles and we'll have to resolve them. And that will always be a process of using arithmetic and maybe some clever ideas.
I want to just quickly go through some of the basic arithmetical steps that we will be using constantly as ongoing recurring themes throughout all that we're going to do together.
Let me begin with just a very simple example to get us back in the spirit of things. Suppose that I have something like 7(3 + 2). Now remember, if there's no sign given in between two numbers like this or two quantities that means that it's understood that it's multiplication. So multiplication is the naked operation. If there's nothing there, you're thinking multiplication, and if there's something there, then you're thinking whatever the "something" says. Now in this case, what would this equal? Well, what I'm going to do is actually focus myself on these parentheses and compute that thing in there because those parentheses are saying, "Do me first." If I do that first, what I see is 7 x 5 and that equals - well, I can do that. That's 35.
So what's the big deal? Well, the big deal is that sometimes in life we're not going to actually want to compute this number this way. Let me show you a different way of thinking about it. Another way of thinking about it is to first consider this multiplication, the naked sign right here. Now if that 7 is going to multiply this entire quantity then I have to make sure that 7 hits every single person that's over here. So what I have to actually do is make sure that I have 7 hitting that 3 and then 7 hitting that 2 and so what I'm seeing actually is the following: 7 x 3 - sometimes they use a dot for times, by the way. Do you see why? If I didn't, it could look like 7 naked 3 or it could look like 73. Mathematicians went crazy for about 100 years before they said, "Why don't we just put a dot there?" They said, "OK. That takes care of that problem." Then I have to take 7 and multiply it by the 2. You see, I've got to take 7 and multiply it by everybody. Now what happens when I do that? When I do that, this gives me what? Well, 7 x 3 = 21, then plus 14 and this still gives me 35. Of course we get the same answer for doing it correctly but this illustrates an important property of the numbers and this important property is called "The Distributive Law." The distributive law, where if you have something multiplying a quantity where inside that quantity you have things that are being added or things that are being subtracted, then if you want to distribute that outside thing, you want to make sure it hits every single person and we will use this time and time again. This is a real big one. OK. Great. That's a fun one.
Let me do another example just to illustrate the distributive law to you and try to make it a little bit more abstract. So how about 5(3a + 4b)? Now when we see a's and b's and x's and y's and z's and w's, and if you're really fancy maybe you see an alpha or a beta or something like that, don't even panic for a second. All that I want you to do is to realize that that's representing some number and I just haven't told you what it is. You could think of it as 4. You could think about it as -3. I don't care, but just think about it as a number and if you do that, you can't go wrong. For example, here if you use the convention I just said, this is 3 naked a. That means 3 x a. Whatever a is I'm multiplying it by 3. That's all this means.
Now to use the distributive law, I can take the 5 and multiply it by this term and multiply it by that term. I have to make sure it hits every term that's being added or subtracted. In this case there's two terms and they're being added together. When I do that, what do I see? I see 5 x 3a so that's 15a plus 5 x 4b, which is 20b. Do you see how I took the 5 and multiplied it by this term and then added it to 5 multiplied by that term? Now these can't be combined at all because of course we don't know that a and b are the same. They might be different. In fact, all I can do is just write that. There's the distributive law in action.
Now there's some other fun things I wanted to share with you. For example, there's always something great that mathematicians love to do. When you don't know what to do in life, do you know what you should do? Do nothing and, in fact, we're going to be doing nothing a lot in this course. In fact, most of this course is, in fact, doing nothing and that is really at the heart of algebra. That didn't come out too well but, the point is doing nothing is a very valuable and important idea and let me show you how you can do nothing. Suppose I have an equation. For example, x = 3. One way I can do nothing is to add zero to both sides. You see if I add zero to both sides, adding zero does nothing. I haven't done anything. In fact, one way I'm doing nothing is to take a number and just add zero. That's because zero is thehere's the technical wordthe additive identity. It's the number that when you add it to anything else you just stay the anything else. That's doing nothing. Now you can do this with respect to multiplication. If you do it with respect to multiplication, what do you think you do? You multiply it by some number that's not going to change the value. That's the number 1. And so you have, for example, 7 x 1 and that equals 7. Doing nothing is something that's going to be very important to us, and you'll see why when we're trying to actually solve equations. We'll want to do nothing all over the place and you'll see that in action.
Another thing I wanted to tell you is how you can actually untangle things. For example, suppose I have something like this. Pick a number. Think of your favorite number. Was it 3? We have 3. Now suppose I want to add something to 3 to make that equal zero. What should I add? Well, what we add is what's called the additive inverse and it's just the negative of 3 or also known as -3. In fact, the additive inverse is just the negative of a number. Now you've got to be a little bit careful because what's the additive inverse, and I'll talk about this in a bit, of -4? What do you have to add to -4 to make it zero? Well, here it's actually 4 or negative negative 4. An additive inverse is the number you have to add to somebody to make the result equal to zero.
You can do this with respect to multiplication, too. You can talk about multiplicative inverses. For example, 7, what do I have to multiply 7 by in order to make it equal to 1? Well, the answer is. It's the reciprocal. That's actually not bad at all. For example, suppose I had. What do I have to multiply by in order to make it equal to 1? That's going to be the multiplicative inverse, which in this case would be 2. That was pretty easy to find as well.
Now the operations of multiplication and addition satisfy a lot of great properties. I want to tell about some of them really fast because it'll be useful. For example, they're commutative. That means it doesn't make a difference if you have 7 + 3 or 3 + 7. It always gives you the same answer. Let me show you. 7 + 3, well that equals 10, but if I did it in the other order, 3 + 7, that still equals 10. This is also true with multiplication. Take (2)(5). That equals 10 and that's the same thing as (5)(2).
Order does not make a difference. If you have two things and you're adding them together, it doesn't make a difference how you do the operation. That's called commutative and that's going to be a real fun one. That's my favorite.
Now there's also a property called associativity. Now associativity is a cool one because what associativity says is if I have a whole bunch of plus signs there, add it up any way you want. Let me show you what I mean. Suppose I wrote this: 1 + 2 + 3. What does that mean? Does that mean first take 2 and 3, add them together, and then add 1, or does it mean take 1 and 2, add them together, and then add 3? Associativity says, "It's your choice. Whatever you want is great." Because it says that if you want to group it like this, that's the exact same way as first doing the first two terms and grouping like this. Let's check. 2 + 3, if I add them together I get 5, and that gives me then a 6, and on this side here I have a 3 + 3, and even though I have a different intermediate step, all correct roads lead to the correct answer. That's what associativity is.
And that's also true with multiplication. If I have 2(3)(4), it doesn't make a difference what order I do it in. I could say 2 times 3, which is 6, times 4 is 24 or I can say 3 times 4, which is 12, times 2 is 24. Associativity is a really great one.
Now I want to end with telling you about one of my favorite things in the world, zero. I love zero. I'm going to be coming back to zero because this is all I've got is basically zero. Zero is great because if you take any number - tell me your favorite number. Was it 17? I knew it. If you multiply it by zero, you get zero and this is true no matter what the number you put here is. Any number you want, if you want to kill it off, just multiply it by zero and it will all become zero. It's fantastic. There's no other number like zero.
Now there's a little downside to this, though. There's a downside and the downside is there will be no dividing by zero. That's right. There will be no dividing by zero, and let me show you why. Suppose I take 14 and I divide it by 2. Well, if you think about that, that's 7. And why is it 7? It's 7 because 2 x 7 = 14. Now what if I took a number like 14 and tried to divide it by zero, then what would that equal? Well, it would equal question mark. What's the property of question mark? Well, question mark times zero has to equal 14, but wait a minute. I just told you that anything times zero is zero. There is no question mark that I can multiply by zero to get 14. You can't divide by zero. This is a big no-no. You can't take any number at all and divide it by zero. You are breaking the laws of algebra. So as long as you avoid that, you could do whatever want, but there will be no dividing by zero. Multiplying by zero, great. Dividing by zero, you're going to get the claw.
I'll see you at the next lecture.
The Real Numbers
Properties of Real Numbers
Properties, Identities, and Inverses Page [3 of 3]

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