# Distributive property-Subtraction

1. ### C0nfused

139
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. ### arildno

12,015
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)

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