Distributive property-Subtraction

  1. Hi everybody,
    1) We have defined the distributive propery of multiplication like this:
    a(b+c)=ab+ac and (a+b)c=ac+bc . So when we have (a+b)(c+d) , how do we get the result using the above definition? We just consider one of the parentheses as one number so we get (a+b)c+(a+b)d for example(we think of (a+b) as a number g?)?

    2) And one more thing: we define -x as the number that when added to x gives a sum 0. We also define that -x=(-1)x and a-b=a+(-b) (definition od subtraction). So when we have an expression like this: a-b-c+d-e this is considered a sum ? I mean the minus signs in the above expression show subtraction or the above is the same (i mean not only in the result but also in the interpretation of it) as this: a+(-b)+(-c)+d+(-e) ?

    The 1st refers to multiplication of reals or generally for scalar multiplication in a vector space or multiplication in a field
    The 2nd refers to reals but also generally to addition in a vector space

    They may be silly questions but i like to understand things by using only the definitions

    Thanks
     
  2. jcsd
  3. arildno

    arildno 12,015
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    Gold Member

    You're right about 1)
    As for 2)
    We do NOT define -x=(-1)*x, we prove that statement as follows:
    a) For any real number "a", we have a*0=0
    PROOF:
    z=a*0=a*(0+0)=a*0+a*0=z+z, that is: z=z+z
    But, since "z" is a real number, it has an additive inverse -z:
    z+(-z)=z+z+(-z) which means 0=z.
    which was what we should prove.
    b) The additive inverse of a number is unique:
    Proof:
    Suppose z2 was an additive inverse to z other than (-z).
    Then:
    0=z+z2, adding (-z) to both sides yields:
    (-z)=z2

    c) Since x=1*x, we have:
    x+(-1)*x=x*1+x*(-1)=x*(1+(-1))=x*0=0, by a).
    Bot from b), it then follows that (-1)*x=(-x)
     
  4. Thanks for your answer. I think that one part was not answered:

    "So when we have an expression like this: a-b-c+d-e this is considered a sum ? I mean the minus signs in the above expression show subtraction or the above is the same (i mean not only in the result but also in the interpretation of it) as this: a+(-b)+(-c)+d+(-e) ?"
     
  5. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    Oh, yes:
    In this perspective, there exists only two operations: Multiplication and addition.
    The subtraction a-b is a short-hand notation for the addition a+(-b)
     
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