The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

The Commutative Property Around Us

Date: 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 


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


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   

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 

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

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.
Associated Topics:
Elementary Addition
Elementary Definitions
Elementary Multiplication

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.