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Beg Algebra: Multiplying and Dividing Decimals

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  • Type: Video Tutorial
  • Length: 7:59
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 85 MB
  • Posted: 06/27/2009

This lesson is part of the following series:

Beg Algebra: Decimals (4 lessons, $5.94)

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 and Dividing Decimals
Okay. So adding and subtracting decimal numbers are no big deal. The important thing there is to, remember, combine--add or subtract--like terms. So there we want to line up those decimal points just right and then let everything fall down. Let the digits fall where they may. So that's great.
Now multiplying and dividing is a different can of worms because there, what we've got to do is remember that the decimal point represents actually a placement of how many times we're multiplying by some power of 10--so either 10, 10^2, 10^3, and so on. So that's going to inform how we do multiplication a little bit differently. So let me show you how multiplication and division go with decimal numbers. Let's suppose we want to multiply 5 point--which I'm going to denote by one of these mints, this is one of these, I think, sophomore mints--5.31, and I want to multiply that by the number 2.5. Now notice that this is a dramatic departure from the adding and subtracting ideas we talked about before. There, we can only add and subtract like terms, but with multiplication, we can actually do things that look like this. In particular, what we do is we actually just literally line them up and in fact what I'm really going to do is pretend for the moment the decimals aren't there. Now if we think about them in whole numbers, we can just multiply those through the way we saw how to multiply numbers. So I'm actually going to do that. That is to say, I know that they're there, but really, I'm going to pretend they're not. So I don't know if you'd like me to not keep them there and do the multiplication or keep them there. I don't know. What do you think? I think we'll keep them there, but we'll understand that they'll be blinking like this. I'll put them real low so they don't get in the way. Now let's just do the multiplication of those two numbers. How do you do that? Well, we know how to do that. We first take the 5 and multiply it by every single term here. Then we slide over and do the same thing with the 2. So 5 o 1 = 5. 5 o 3 = 15, so I write down the 5 and I carry the 1 over here, and then here I have 5 o 5 = 25 + 1 = 26. Now I come to the 2, I slide over. 2 o 1 = 2. 2 o 5 is--I'm sorry, 2 o 3 = 6, and 2 o 5 = 10. Notice how carefully I am to line up everything. Boy, if someone is sloppy here, that's fatal. Now I add these two answers together. Remember, there's an invisible 0 here so that's just a 5 + 2 = 7, 6 + 6 = 2, carry a 1 because it's actually a 12, 2 and the invisible 1 is a 3, and I have the 1. That is the correct answer to the product 531 o 2.5, but that wasn't the correct answer, so there's going to be a decimal here, and now how do we actually figure out where to put the decimal? Well, the way we think about it is you can take the sophomore mint and just slide it over to the right and every place you slide over costs you a slide here to the left. So let's try it now and see what happens. So if I slide this over, it forces me to do that. So now I'm here. So let's slide this over. It comes to here and slide this over and it comes to here and that's the final answer. The final answer is 13.275. How did we get there? We just--for every move of the decimal, one, two, three, we move this number; 1, 2, 3. Does the answer make sense? This is around a little bit bigger than 5, this is a little bit bigger than 2. What's a number that's a little bigger than 5 multiplied by a number that's a little bit bigger than 2? The answer is a number a little bit bigger than 10. It checks. If you got an answer like this; 132, or you got an answer like this; .13. If you just think about the answer, and it doesn't make sense, you immediately see you made some mistake, either in the multiplication or in the moving the decimal over, because this is around 5, this is around 2, you multiply them together and you should get something around 10. So it should be here, so that confirms with the idea of moving 1, 2, 3 over. We move 1, 2, 3 over here. No problem, no problem.
Okay. Let's try a division one really fast together. Let's take--and division, as you all know, of course, is brown. You know that don't you? I just made that up by the way. Okay. Here we go. So let's take a look at 1.2 ÷ 3. Well, how would you do this? Well, the way--there's several ways of doing this. One way is just to write it as a division, as a fraction, so we'd have 1.2/3. Remember, these two things are the same. 1.2 ÷ 3, we saw that written as 1.2/3, and now we can do the same thing--ooh, I should have used a sophomore mint. Wait a minute, I'll fix it right now. Watch this. This is live, right there on your computer, live. It's a white thing, and now with the white thing there I can put a sophomore mint right there and what I want to do is I want to have all the decimals to the far right. So I pick this up and I want to move it, but that's just to multiply by 10. So if I'm going to multiply by 10, then I've got to multiply the bottom by 10, and if I multiply the bottom my 10, that this moves over and so now I have a 30. So notice that 1.2/3, if I multiply the top by 10 and the bottom by 10, that's just multiplying by 1, here the decimal shifts over, and I get a 12, and here I get a 30. So, again, it's the same shifting process, and now I can actually reduced this a little bit. Let's reduce this. So here I see that there is a 6 factor on the top. This is actually 6 o 2. On the bottom, I see this is actually 6 o 5. So I can actually remove the common factor, and I see 2/5, and now that's the answer. That is actually what 1.2 ÷ 3 is, or if you don't like that, and you want that in decimal, what do you do? You just do a long division really fast. 5 ÷ 2 point--and now we put a decimal point--5 doesn't go into 2, so that's a 0. 0 o 5 is 0, I subtract, I get a 2. I now bring down the next term and 5 goes into 20 four times, and so I see, in decimal form, .4 or maybe 0.4 if you'd like.
The point is, all you've got to do is just do the multiplication or do the division and then just move the decimal over accordingly. Not a big deal, just careful with your little sophomore mints. They come in handy don't they? See you at the next lecture.
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