1. ## Number system

The rational system that in the picture was wrong (the blue rectangle in picture) because the solution is:
2/3 + 3/4 = (2*4)/(3*4) + (3*3)/(3*4) = 8/12 + 9/12 = 17/12
Why the solution of complex system in the yellow rectangle is wrong?

2. ## Re: Number system

Originally Posted by policer
The rational system that in the picture was wrong (the blue rectangle in picture) because the solution is:
2/3 + 3/4 = (2*4)/(3*4) + (3*3)/(3*4) = 8/12 + 9/12 = 17/12
Why the solution of complex system in the yellow rectangle is wrong?
You are not showing us the whole problem. Namely, you have not defined multiplication on ordered pairs. Is multiplication on ordered pairs $(a,b)\cdot (c,d) = (ac,bd)$? Because if it is, then what you have in the yellow box is correct. If it is not defined that way, then what you have in the yellow box may be wrong. But, without any definition, we have no way of judging.

3. ## Re: Number system

The question is a big question:
"Why complex number isn't defined by this way?"
One of the part on the answer is "because the definition of (a, 0) and (0, b) isn't defined."
O.K. I don't remember the formula of product of complex number.
And, why the (a, 0) and (0, b) aren't have inverse.

4. ## Re: Number system

Originally Posted by policer
The question is a big question:
"Why complex number isn't defined by this way?"
One of the part on the answer is "because the definition of (a, 0) and (0, b) isn't defined."
O.K. I don't remember the formula of product of complex number.
And, why the (a, 0) and (0, b) aren't have inverse.
In the usual complex field $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$

5. ## Re: Number system

So, the solution is:
(a,b) * (c, d) = (ac - bd, ad + bc)
(2,3) * (3,2) = (2*3 - 3*2, 2*2+3*3) = (6 - 6, 4+9) = (0, 13)
Right?

6. ## Re: Number system

Originally Posted by policer
So, the solution is:
(a,b) * (c, d) = (ac - bd, ad + bc)
(2,3) * (3,2) = (2*3 - 3*2, 2*2+3*3) = (6 - 6, 4+9) = (0, 13)
Right?
Frankly, I do not understand what it is that you are asking. Why not start over from the beginning and lay out the exact problem?

7. ## Re: Number system

Originally Posted by policer
Why the solution of complex system in the yellow rectangle is wrong?
\begin{align*}(2,3) \cdot (3,2) &= (2+3i )(3+2i) \\ &= 6 + 9i + 4i - 6\end{align*}

8. ## Re: Number system

Originally Posted by policer
The rational system that in the picture was wrong (the blue rectangle in picture) because the solution is:
2/3 + 3/4 = (2*4)/(3*4) + (3*3)/(3*4) = 8/12 + 9/12 = 17/12
Why the solution of complex system in the yellow rectangle is wrong?
We are still completely lost on what it is you mean. There are a number of possibilities for (a, b) x (c, d). We can't answer your question until you tell us what it is.

-Dan

9. ## Re: Number system

Originally Posted by policer
So, the solution is:
(a,b) * (c, d) = (ac - bd, ad + bc)
(2,3) * (3,2) = (2*3 - 3*2, 2*2+3*3) = (6 - 6, 4+9) = (0, 13)
Right?
Yes.

10. ## Re: Number system

I have an Hebrew article that say because:
(0, a) and (b, 0) has no inverse the notation (a, b) * (c ,d) isn't (a*c, b*d).
i will give you more detail if you want.

11. ## Re: Number system

Originally Posted by policer
So, the solution is:
(a,b) * (c, d) = (ac - bd, ad + bc)
(2,3) * (3,2) = (2*3 - 3*2, 2*2+3*3) = (6 - 6, 4+9) = (0, 13)
Right?