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About this Lesson
 Type: Video Tutorial
 Length: 7:50
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 83 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, 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 UTAustin 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, padic 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.
About this Author
 Thinkwell
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11/13/2008
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Real Numbers
So, we've been thinking about lots of different types of numbers, and I wanted to share with you sort of a global sense of what a number is. Really, all the numbers that we have been thinking about up to this point are all examples of what we call "real numbers." This is great. Real numbers, here we go, this is the real numbers, it's not the fake numbers, it's not the sort of the wimpy numbers. This is it, this is really what we're trying to shoot for. So what are the real numbers? Well, all the fractions that we thought of are examples of real numbers. And, in fact, all the whole numbers are examples of real numbers. And, in fact, there are lots of others. So when you think of the real numbers, what should we be thinking about? Well, the answer is that a real number is a point on the real number line. So let me remind you of what the real number line is. We've seen this in elementary school, you know, sometimes on our desks we had this on there, right? You know, maybe 0, 1, 2, 3 and so on. If you went to like a private, fancy school, you had the negative numbers there. But anyway, there they all are. And what does that mean? What does that mean exactly? Well, every single point, every single location represents a real number. So what does that mean? Well, first of all, you'll notice that we have the regular counting numbers. So, those are the numbers 1, 2, 3, 4, 5, 6 and so on. Those are called the counting numbers, because that's what we used to count, or sometimes called the natural numbers. If you include the number 00, 1, 2, 3, 4, 5, 6you give the possibility that, in fact, you might have nothing, those are actually called the whole numbers. So, the whole numbers are just the numbers 0, 1, 2, 3, 4, etc. Now, if you're allowed to go into deficit spending, then you can actually get the set of integers. The set of integers include the whole numbers, but also all the negative reflections. So, in fact, the integers start off at negative, you know, way out there, negative infinity or something, and then all the way up to negative 6, negative 5, negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, 4, 5 an so on. Those form the next bigger collection of objects, which are called the integers. Okay, but it doesn't stop there, because we've already seen the fractions. Now the fractions, of course, we know, well, we can think of all of these numbers that we've seen so far, integers, as fractions, because it's minus 6 divided by 1, so that's okay. But, in fact, there are a lot more, right? For example, we have a . Where's ? Well, in fact, you can think about in terms of a location. Now what's the location? Well, would be located right strictly in between the 0 and the 1. So it would be right here. There's the location of . So the fractions are peppered all the way through the real number line. And this peppering through the number line actually provides sort of all dots, dots, dots, and it may seem as though you put down all those dots, you'll get the complete number line. In fact, in ancient Greece, people actually believed that. They thought that every real number, in fact, was a fraction, was what's called a rational number; ratios, rational, fractions. But it turns out that was wrong, and one of the greatest triumphs of humankind was the discovery of numbers that in fact are not fractions. They are not rational, and they are called irrational numbers. So there's a whole other side, the dark side, if you will, of the real line, which are the irrational numbers. Now, you all know irrational numbers, we know them, you and I know them, because, for example, the square root of 2 is an example of an irrational number, although we haven't seen why, but it turns out it is. The number pi, used with circles, is another example. And there are lots. In fact, there are tons of examples, and you can really visualize the number line as a place to locate things. So, for example, let's take the square root of 2. If you take a look at the square root of 2, you can compute it on a calculator. Let me show you how you would actually do that, by the way. You just take a regular calculator like this. I don't know if you can see that are not, here I have a little magnifying glass, so here you can maybe see it a little bit better. And I'm just going to put in 2 and hit the square root key. And do you see what we get? We get 1.41...can you read that? I can't focus it. There we go, 1.4121... and it goes on. Here, I'll show you what it says. You can try it on your own calculator: 1.414213...And this is producing the beginning of an infinitely long decimal expansion. So this number has a decimal expansion. Every real number, in fact, has a decimal expansion. We talked about decimal expansions for fractions already, but now you're seeing them, in fact, for any real number. Now, what does that decimal expansion imply in terms of our picture? Well, it turns out you can think of the number line as like an infinitely long street and each number is the street address where that number lives. So, for example, if you want to visit the square root of 2, a perennial favorite, what would you do? Well, you would first, again, divide and conquer, one of the major themes of this course. Just don't do it all at once, don't point to the square root of 2. Start off by saying it's 1 point something. That means it lives between 1 and 2. The integer part of this is 1, the whole number part, so it lives somewhere in between here and here. So, the square root of 2 resides somewhere in here. You can get rid of everything else, and what we do is we actually do that. We get rid of everything else. And if we enlarge that, if we now take the magnifying glass, and enlarge and look really closely between 1 and 2, what you see is a little number line, a little number line segment, and if you enlarge that to show detail, here's what it looks like. So, here's the 1, and here's the 2. And in between there, when we look a little bit closer, all the sudden we see that we get the next decimal spot. So, we get 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2. So, the next decimal digit, the .4 here, that 4, tells me which subneighborhood between the 1 and the 2 that the square root of 2 resides, and we see it's a 4. So, we actually go right here, 1.4, it's in between 1.4 and 1.5. So, originally we just knew it lives somewhere in here. We enlarge that, and what do we see? We see now that it lives somewhere in between here. What do you think we do now? That's right. We enlarge this thing. We take the magnifying glass and enlarge between 1.4, where are you 1.4, 1.4 and 1.5, we enlarge that line segment right there, and what do we see? The further, we go, the more decimal digits we have. And, so now this little segment right here is enlarged to show detail, 1.4 and 1.5, and we get the next digit of accuracy, and this tells me that I'm living in between, well, 41, so here's 41 and 42, so it's in here. And, if you follow this back, that's a little teeny thing right here, which is a little teeny thing back here, and you see that the further out you go, the more you're honing in on the real number. That gives a sense of the connection between the decimal expansion, which is a numerical thing, and the visual notion of real numbers as points on a number line. As you go further and further out, we're cutting things into finer and finer and finer mesh, and if you go out forever, it's like infinite directions to locate and pinpoint precisely that one point, whether it's a fractiona rational number, or not a fractionan irrational number. So, the integers are depicted numerically here. But, in between the cracks, we've got a jungle of real numbers.
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