Distributive property-Subtraction

  • Thread starter C0nfused
  • Start date
  • #1
139
0
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
 

Answers and Replies

  • #2
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
134
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)
 
  • #3
139
0
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) ?"
 
  • #4
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
134
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)
 

Related Threads on Distributive property-Subtraction

  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
4
Views
15K
Replies
2
Views
1K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
20
Views
15K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
4
Views
2K
Replies
4
Views
5K
Top