i don't understand this notation

C

cotfw

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I was reading through a book and it wrote this:

(a,b) + (c,d) = (a+c,b+d)

I don't understand this though. How can number pairs be added together as if they are values
 
cotfw said:
I was reading through a book and it wrote this:

(a,b) + (c,d) = (a+c,b+d)

I don't understand this though. How can number pairs be added together as if they are values
Click to expand...

It is simply a matter of definitions.

Define multiplication as: \(\displaystyle (a,b) \cdot (c,d) = (ac - bd,ad + bc)\)
 
cotfw said:
I still don't understand
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Those two operations are absolutely standard. They are the ways we define addition and multiplication in the field of the complex numbers. You may not be mathematically mature enough to appreciate the definitions.
 
pka is assuming that your (a, b) represents the complex number a+ bi. We add complex numbers by adding the real parts and the imaginary parts: (a+ bi)+ (c+ di)= (a+ c)+ (b+ d)i. Using (a, b) to represent a+ bi and (c, d) to represent c+ di, that would be the same as (a, b)+ (c, d)= (a+ c, b+ d).

But perhaps you intend (a, b) to represent the vector ai+ bj, a vector with component along the x-axis equal to a and component along the y-axis equal to b (here, i is the unit vector in the x-direction, j the unit vector in the y-direction). In that case, we add "component wise", adding the x components together and adding the y components together: (a+ bi)+ (c+ di)= (a+ c)+ (b+d)ix. Again, if we represent a+ bi by (a, b) and c+ di by (c, d), that is the same as (a, b)+ (c, d)= (a+ b, c+ d).

The only real difference between "complex numbers" and "two dimensional vectors" is that there is a product defined on complex numbers that is not defined on vectors.
 
HallsofIvy said:
pka is assuming that your (a, b) represents the complex number a+ bi. We add complex numbers by adding the real parts and the imaginary parts: (a+ bi)+ (c+ di)= (a+ c)+ (b+ d)i. Using (a, b) to represent a+ bi and (c, d) to represent c+ di, that would be the same as (a, b)+ (c, d)= (a+ c, b+ d).

But perhaps you intend (a, b) to represent the vector ai+ bj, a vector with component along the x-axis equal to a and component along the y-axis equal to b (here, i is the unit vector in the x-direction, j the unit vector in the y-direction). In that case, we add "component wise", adding the x components together and adding the y components together: (a+ bi)+ (c+ di)= (a+ c)+ (b+d)ix. Again, if we represent a+ bi by (a, b) and c+ di by (c, d), that is the same as (a, b)+ (c, d)= (a+ b, c+ d).

The only real difference between "complex numbers" and "two dimensional vectors" is that there is a product defined on complex numbers that is not defined on vectors.
Click to expand...

Thank you that was very helpful
 
HallsofIvy said:
pka is assuming that your (a, b) represents the complex number a+ bi. We add complex numbers by adding the real parts and the imaginary parts: (a+ bi)+ (c+ di)= (a+ c)+ (b+ d)i. Using (a, b) to represent a+ bi and (c, d) to represent c+ di, that would be the same as (a, b)+ (c, d)= (a+ c, b+ d).

But perhaps you intend (a, b) to represent the vector ai+ bj, a vector with component along the x-axis equal to a and component along the y-axis equal to b (here, i is the unit vector in the x-direction, j the unit vector in the y-direction). In that case, we add "component wise", adding the x components together and adding the y components together: (a+ bi)+ (c+ di)= (a+ c)+ (b+d)ix. Again, if we represent a+ bi by (a, b) and c+ di by (c, d), that is the same as (a, b)+ (c, d)= (a+ b, c+ d).

The only real difference between "complex numbers" and "two dimensional vectors" is that there is a product defined on complex numbers that is not defined on vectors.
Click to expand...

Just to add to the discussion (not to contradict anything):

However, vectors have two types of vector-products defined.

Though, vector division is not defined.
 
Since the original post and the previous response said nothing about "multiplication" or "division" I don't see how that "adds" anything to the discussion.
 
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