Toronto Math Forum
MAT3342020F => MAT334Lectures & Home Assignments => Chapter 1 => Topic started by: Zhekai Pang on September 19, 2020, 11:17:42 PM

http://forum.math.toronto.edu/index.php?topic=1278.0
Why is it a whole line? I also think Re$(z) \geq 0$ is required because the original equation is Re$(z) =  \cdot$.

Reply to Zhekai:
I think it is explained by the first reply in that link. x+yii=Rez is equivalent to x^2 + (y1)^2=x^2, subtract both sides by x^2 and this equation would be irrelevant to x. So the only restriction is on y and we have to let y=1. Also, if put it in geometry, it does not matter whether Rez has a positive sign or a negative sign since x+yii means that the distance between (x,y) and (0,1) is fixed and equal to Rez=x, and it has two corresponding points, (x,1) and (x,1).

I disagree. By definition of norm, it is nonnegative. Thus, in order for Re$(z) = \cdot$, it has to be nonnegative. You cannot make the first equivalence. For example, $z = 3$ is an empty set, although $x^2+y^2=9$ is a circle.

halfline