The Commutative Property Around UsDate: 11/24/98 at 15:28:25 From: Tabitha Leber Subject: The commutative property around us Our teacher handed all of us a question sheet today before we left class. I am going to a new school this year and have never heard of the commutative property. I can't find it in the math book. I can't find it on the computer. I don't know where to look for it. My parents have never heard of it. Here are a few examples of what I'm suppose to answer: WHICH OF THE OPERATIONS BELOW ARE COMMUTATIVE AND WHICH ARE NOT? EXPLAIN YOUR ANSWERS 1. To put on your coat and to pick up your boots 2. To wash your clothes and to dry them 3. To put on your left shoe and to put on your right shoe 4. To hang up the phone and to say goodbye Thanks, Tabitha Date: 12/02/98 at 15:27:48 From: Doctor Gail Subject: Re: The commutative property around us Dear Tabitha, I must tell you first that I love your teacher and I think it is important for you to know why. What your teacher is really trying to get you to do is to connect the meaning of the commutative property to the world around you. So let me help. The commutative property of addition says that changing the order of addition of two numbers does not change the meaning. For example, 2+3 = 3+2. For multiplication, 4*5 = 5*4. There is no commutative property for subtraction, since it doesn't make sense to think that 10-5 = 5-10. There's a difference between having $10 and spending $5, and having $5 and trying to spend $10, so subtraction is not commutative. Now back to the questions from your teacher - but I'm going to present one of my own. Consider putting on toothpaste and brushing your teeth. Are the results the same if you put on toothpaste, then brush, as compared to brush and THEN put on toothpaste? I think one version will make the dentist sad! Let's consider someone who likes sugar and cream in her coffee. Does it make any difference if the sugar goes in followed by the cream, or if you put the cream in first followed by the sugar? I think the results are the same to the coffee drinker, so I think the sugar and cream example is commutative. I hope this helps! Write back if you have more questions. - Doctor Gail, The Math Forum http://mathforum.org/dr.math/ Date: 03/05/2003 at 11:12:06 From: TimeForLime Subject: A polite correction Preparing coffee is NOT necessarily COMMUTATIVE. A New Yorker, say, wouldn't understand your example at all. Purchased coffee, say "commuter coffee," sold one cup at a time in containers "to go," is often only marginally warm enough. If you add cream first and then sugar, the sugar doesn't always dissolve thoroughly, or at least as quickly. You HAVE to dissolve the sugar FIRST. So it isn't COMMUTATIVE in the purest sense. If you have always obtained hot, hot coffee from an urn inside a warm faculty building or teaching facilty, you might not be aware of this. Thanks for listening. Date: 03/05/2003 at 13:15:29 From: Doctor Peterson Subject: Re: A polite correction Hi, TimeForLime. I suppose a commuter ought to have the last word on commutativity! ;-) Actually there is a serious point to be made here: unlike math, anything we say about the real world is likely to be false under some circumstances, because we never know all there is to know. In math, we can make absolute statements because we are defining the complete circumstances - we know that numbers do not behave differently when they are cold, because we define them to be independent of temperature. Anything we say about coffee is conditional on what coffee we are talking about. But if we say that the coffee is hot, maybe the illustration can stand. There is, in fact, a similar situation with numbers: In a computer addition might not be commutative, because the numbers might be stored in different-sized variables. If you add a large number to a number stored in a small space, there might be an overflow, which would not happen if you added the small number to the large one. Again, what's happening is that mathematical numbers do not accurately model the computer's variables. Thanks for the laugh, and the chance to think! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 03/10/2003 at 05:11:48 From: TimeForLime Subject: Thank you (A polite correction) Thank you for taking my comment seriously. Yes, the business of taking "math" numbers in different order used to happen on the slide rule, before the computer. Your point that adding, say "single precision" to "double precision" - as an example of "numbers stored in different sized variables" - is not necessarily commutative is well taken. I'm satisfied. Case closed. I enjoy all your examples. I just think examples like the toothpase tube is SO unequivocal that it's probably not necessary to tempt bright students (I'm not, and wasn't then) with marginal examples like coffee. I'm sure your genius students could even find fault with the toothpaste tube but they might have to position themselves near a singularity to pull it off. Now we've both had a laugh. Bye. |
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