1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Can commutativity of multiplication and addition under real numbers be assumed?

  1. Sep 27, 2010 #1
    1. The problem statement, all variables and given/known data

    so.. let the operation * be defined as x*y = x + y + xy for every x,y ∈ S,
    where S = {x ∈ R : x ≠ -1}. Now i have proven associativity, existence of an identity and inverses, all without commutativity, but i must show that this is an abelian group, so now i have to show commutativity. I know that multiplication is not always commutative, but i am not using matrices, just reals/{-1}.

    2. Relevant equations

    the operation * is defined by x*y = x + y + xy.

    3. The attempt at a solution

    i simply say that first, addition is commutative on the reals, so x+y=y+x, by the commutative law of addition of real numbers, and second that xy=yx, by the commutative law of multiplication of real numbers. this would show that x*y=y*x. Is this allowed or do i have to show commutativity without assuming addition and multiplication are commutative on real numbers? and if i have to show commutative without the assumptions above, can someone point me in the right direction on the proof?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 27, 2010 #2
    in case i did not show enough of an attempt at a solution, here is a more detailed attempt summary...

    the operation * is commutative if for every x,y∈{x∈R:x≠ -1} such that x*y = y*x.
    But, x*y = x + y + xy, and
    y*x = y + x + yx,
    = y + x + xy (by commutative law of multiplication of real numbers)
    = x + y + xy (by commutative law of addition of real numbers)
    So x*y = y*x, and thus, * is commutative.

    this is my original solution but i am not entirely sure whether it holds up. it seems like using the law of commutativity of multiplication/addition of real numbers might be too "flimsy" to use in this kind of proof. it would sure be great if i could hear some feedback.
  4. Sep 27, 2010 #3
    You are correct in your reasoning. If x*y = x+y + xy, then y*x = x+y +yx, which holds since addition and multiplication are commutative in the reals.
  5. Sep 27, 2010 #4
    so it is not necessary to prove commutativity of addition or multiplication in real numbers here? just making sure...
  6. Sep 27, 2010 #5


    User Avatar
    Science Advisor

    No, you do not have to "reinvent the wheel" for every problem. You do not have to prove the properties, that have already been proven, for the "usual" operations on the real numbers. if you are given some new operation, defined using the "usual" operations, you can assume all of the properties of the "usual" operations in order to prove (or disprove) those properites for the new operation.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - commutativity multiplication addition Date
Prove a statement using Peano's Axioms Mar 5, 2018
Prove that multiplication is commutative Aug 10, 2012
NxN matrix multiplication; commutativity Jan 20, 2011
When is matrix multiplication commutative Sep 15, 2009
Commutative multiplication Sep 12, 2009