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Algebra - Associativity and Commutativity

  1. Aug 7, 2012 #1
    Hey guys and garls. I've recently purchased Serge Langs 'Basic Mathematics' and I'm embarrassed to say im not finding it all that basic! Could someone give me a solid discription on how to find Associativity and Commutativity in basic problems so i can progress!

    If I put a basic problem up could someone step by step take me through how they're solving the problem? I'm more than gutted to get stuck on the first freaking set of problems!

    Heres one of the exercises : (a - b) + (c - d) = -(b + d) + (a + c)
    Associativity = ?
    Commutativity = ?

    Hope somebody can link me to a good resource or talk me through this problem!
  2. jcsd
  3. Aug 7, 2012 #2


    Staff: Mentor

    associativity on addition means you can order how you add things so that (a+b)+c is the same as a+(b+c)

    commutativity on addition means you can rearrange the a+b is the same b+a

    so now for your problem rewrite it as follows:

    ( a + (-b) ) + ( c + (-d) )

    then apply the rules to get the other side

    = ( (-b) + a ) + ( (-d) + c ) ...
  4. Aug 7, 2012 #3
    I don't really have anything to add to jedis post, but...Basic Mathematics is, in my opinion, one of the best pre-calculus/basic math texts out there. However, it is, as the name implies a basic math text written by a brilliant mathematician; so it can be confusing at times. Lang tends to write in the "language" of mathematics, and is very formal, so it can be hard to follow at times if you're just getting into math. I would recommend using it in conjunction with a workbook-style text that allows you to get away from all the formality associated with Lang's text and practice what you're learning in a more casual/relaxed manner.

    (this post assumes you're just getting into math)
  5. Aug 7, 2012 #4
    I did Math's at school but that was 6 years ago now. And lets just say I didn't give it any effort (which I massively regret!) I did pass with a C though but I've pretty much forgotten it all!

    Could you please recommend a book to work along side with 'basic mathmatics'?

    Thanks to the both of you for your replies!
  6. Aug 7, 2012 #5
    I'm in the same boat as you more or less and would like to get back into studying math just for fun. It seems like me and you both just need a good book to help us lay a good foundation for learning math now and later. Just a friendly bump for a good suggestion!
  7. Aug 7, 2012 #6
    I have just recently finished that book. The title implies that it covers the basic mathematics (math before Calculus), but that does not mean that he doesn't go in-depth; most of the exercises are very strenuous and can take some time, not to mention the decent amount of proofs that you will have to make on your own.

    It will definitely pay off to work through the book, but if you haven't even done math in 6 years, then at the very least I would recommend, like others have mentioned, that you get a complimentary book to go along with it, that is maybe less formal and explains certain concepts better (Lang can be vague sometimes, or wants you to figure out what he did on your own). Unfortunately, I wouldn't know of any books like that, so you might have to look around on your own.

    As for the problem that you were having trouble with, I remember Lang saying in that section that, with the Associative and Commutative properties, you can manipulate an expression like that into any form that you want. All that you need to do is move the expression around using those properties until it matches with the final solution.

    There are multiple ways to get to that final form, so if you really need to practice, maybe go through them several times until you feel like you have found the most concise and simple way to change the expression.

    Also, I think that you were looking at the problem the wrong way. If I remember correctly, you were supposed to change the original expression into the final expression, while notifying the usage of commutativity and associativity in each step. It looks as if you were trying to see how he got from one expression to the next without actually working it out yourself, which would have been much easier.
  8. Aug 8, 2012 #7
    I would take one side and start working with it until the other side pops out, changing only one thing at the time. Any change to parentheses, including taking them out and putting them in, is an associative thing; any change to the order of things is a commutative thing.

    The other thing you need to know for this problem, which may have been mentioned already in the book, is that "subtraction distributes": when stuff marked with parentheses has a subtraction sign in front of it, that means the stuff inside the parentheses is really being subtracted (or added, if it was being subtracted before). Try a few examples out with numbers plugged in and you will see why.

    The way I would start (not necessarily better than any other way):
    (a - b) + (c - d) = -(b + d) + (a + c)
    = - b - d + a + c (Associative)
    = - b + a + c - d (Commutative)
    = - b + a + (c - d) (Associative)
    ...and so on.

    Tip: Don't forget, when moving things around, that the subtraction sign stays with the thing being subtracted. Technically subtraction is not a commutative operation (7-2 is a very different thing from 2-7, for example), but since subtracting is the same as "adding the inverse" (8-3 is the same thing as 8+(-3), for example), we can "commute" subtracted things as long as they stay subtracted.
  9. Aug 8, 2012 #8
    These replies will be a massive help, I'm going to set a couple hours aside tomorrow evening and I'll give it a whirl!
    I would still however like another but with abit more depth, just to get through it all with a better understanding and abit quicker. Does anyone have a suggestion for me and Aking?
  10. Aug 13, 2012 #9
    Is anybody out there?! :tongue:
  11. Aug 13, 2012 #10


    User Avatar
    Gold Member

    As Klausas put it there is the "Distributive" property in play along with associative and commutative properties, without the distributive property the minus sign would not be outside the parenthesis
  12. Aug 13, 2012 #11
    I don't understand the exercise. Associativity and commutativity of what should you verify?

    Edit: I see it now, you should get the expression on the right from the one on the left and say where you used these two properties. Correct?
  13. Aug 13, 2012 #12
    Various people have already helped him out with the problem. What he's looking for now is another book to accompany him with Basic Mathematics that gives more in-depth explanations of the material.

    And yes, you should have seen that you get the expression from the right from the one from the left, seeing as how I've gone through this book already and explicitly stated that that was the goal of the problem.
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