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Beg Algebra: Ordering Real Numbers

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  • Type: Video Tutorial
  • Length: 7:12
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 77 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 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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ORDERING REAL NUMBERS
You know, one of the greatest and most important things about the real numbers is that they all live on a number line, and in particular, they live on a line. What does that mean? What are the implications of that? Well, I mean, it's so natural that we just take it for granted, but mathematicians actually get really, really excited about it, and I'm one of them. Because if things are lined up, then that automatically means that there is some sort of order, and so here, I want us to think about the idea of ordering the real numbers. Now, in fact, we know how to order the real numbers--It's so common to us, so second nature almost that the point is if a number is to the right of another number, then the right number is always larger. The number line is increasing as you move to the right, and it's decreasing as you move to the left. So if I give you two numbers; let's say -3 and 4, you can assert, with great confidence, which one is the larger. You would say the 4 is larger because the 4 is coming to the right compared to -3 when placed on the number line.
Now how do we denote that? Well, symbolically, we use this symbol. Now this symbol can be thought of in two different ways. You can think about this symbol where they're sort of standing on their heads, or this symbol where they're standing on their feet. So they're either doing a handstand or not, and what does that mean. Imagine two quantities here. This expression, first of all, the way you would read it is this is greater than this or if they were doing a headstand, I'd say this is smaller than this, and the picture really illustrates it, right. Take a look. You can always remember this by just looking at the picture. Right, big thing is on the left when the inequality symbol is this way. Little thing is here. So big thing is always greater than little thing. Turn it this way; we have little thing is always less than the big thing. So you could write our expression--well, one way you could write it is -3--how does that compare to 4? Well, we would say that it is less than. If you wanted to write the 4 and the 3 in the other order, you could. 4 and the -3 rather, but then the symbol would go this way. Both of these expressions mean the same thing, and how would you read this if you were reading a child story to a niece or nephew and it had this in it? You would say -3 is less than 4 or you'd say 4 is greater than -3. They both mean the same thing. They both mean that 4 comes to the right of -3 on the number line.
Let's take a look at some other examples. You could easily see that for example -5 is less than 0. So I could write -5 < 0. I could write, for example, that 3 is less than 6. 3 < 6. Now, if I wanted to write it the other way, I absolutely could. For example, notice I could notice that -4 is smaller than 2, but if I wanted to have the 2 first, how would I say it? I'd say 2 is actually greater than -4, and that says that 2 comes to the right of -4. So, this idea of ordering is a very natural one, and a very important one that the real numbers have, and we denote that with this symbol, where whatever is on this side is always the larger than what's on this side. So this is larger than that, and if it were flipped, then you would say that this is smaller than this. This is the larger one here on this side.
Well, also since they're lined up, we can actually do something else which is we can actually measure distances between two points, and this is the idea of an absolute value. Now absolute values are great. Here is how they work first of all. They're so simple because if you put those vertical bars--flank a number, just put them around a number--all you do is make sure that the answer is not negative. So, for example, the |2| is just 2 because 2 is a positive number. |16| is just 16 because it's a positive number. The absolute value of -5 however, well, it turns out |-5| equals 5. We make the answer a positive number. What does the absolute value represent? It represents distance. It's saying what is the distance between -5 and 0, and the distance between -5 and 0 is not -5. Take out a ruler and measure. First of all, if you look at a regular ruler, you'll notice something interesting. It starts at 0. It has no negatives. You can't have negative distance. Distance is only positive. If you don't believe me, take a look at a tape measure, and you see tape measures start at 0. They don't start at negative numbers. So distances are always positive. If you want to measure it, you just take this, put it down and you see you have five units--1, 2, 3, 4, 5. So, in fact, |-5| is 5. The |-17| is 17. It represents the distance to the origin, which is where 0 is located and the point. It's always positive. Distance is always positive.
Now, what about the absolute value of 0. Well, the distance from 0 to itself is 0. So the absolute value could equal 0 in the special case when you plug in 0. If you plug in positive numbers, the absolute value is just the positive number. If you plug in a negative number, you have to actually flip the sign and make it back to being positive. You can actually use absolute values to not only find the distance away from 0, but a distance away from any two points. For example, let's find out what the distance is between, let's say, 5 and 1. What we do is, we take 5 and subtract off 1 and take the absolute value, and if you do that, you would see 4, and notice, in fact, the distance between 1 and 5 is 1, 2, 3, 4. So absolute values, in fact, allow us to compute distances between points, and if we just take the absolute value of a single number, that's the distance between that number and 0. So not only are they lined up beautifully so we have one less than the other, but they also line up beautifully so we can see how far apart they are. That is the power and the beauty of the real numbers. You gotta love them. See you soon.
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