Somebody notice me that the expression in the picture have a solution (a=0) or (b=0) and (a and b [together] > 0)...
My question is why the solution that the expression is the same in the complex number.
How can I should know when the solution of an expression will be the same in real and complex number?
The equation is sqrt(a^2 + b^2) = a + b. (It is not an "expression", but an "equation".)
I don't understand the claimed solution. It is true that the equation requires a=0 or b=0, but what does "(a and b [together] > 0)" mean? If they are
both positive, then the equation is
not true. Or perhaps you mean the solution is that either a=0 and b≥0, or b=0 and a≥0? That makes some sense. But what you quoted doesn't say that.
What do you mean by "
the solution that the expression is the same in the complex number"? This makes no sense grammatically. I suppose you have to mean, "Why is the solution
of the
equation the same over the complex numbers?".
But, more important, if a and b are taken to be complex numbers (that is, not necessarily real), then it is really inappropriate to write the equation at all, as there is no principal square root to be represented by the radical! When working with complex numbers, the only proper meaning for the radical is both roots at once, which is not proper in an equation. For example, sqrt(i) = ±(1 + i)/sqrt(2); neither root can be taken as "the" root, as neither is "positive", and there is no other way to chose a principal root. But we don't write non-functions like this in equations.
So my answer is that the equation can't even be
written in the context of complex numbers. Moreover, the
solution can't say a>0, because comparisons don't apply to complex numbers.
But if I grant that the equation makes some sort of sense over the complex numbers, then the solution is
not really the same as over the reals, because the solution is "either a=0 and b is any
complex number, or b=0 and a is any
complex number". Over the reals, the solution is "either a=0 and b is any
positive real number, or b=0 and a is any
positive real number". These are not the same solution.
Who told you this? Do you really trust them? Much of what you are asking about is nonsense.
Have you solved the equation yourself, or seen a proof? Don't just assume it, do it -- and then try applying the same reasoning to complex numbers, to see the difference.