What does the complex number 1/2*(1+i) represent in linear mapping?

[loy]

Homework Statement

i recently saw a question about complex number, and its answer about the center of a circle is 1/2*(1+i). what does that mean?

Homework Equations

f:ℂ → ℝ^2

The Attempt at a Solution

since we define z=x+yi is an element of ℂ, so by the mapping above,
we can say that f:ℂ→ℝ^2 = f(x+yi)=(x,y)?
and by the inverse function of f,which is f-1(x,y)→(x+yi)?
From the statement above, f-1(1/2,1/2)=1/2+i/2 =1/2*(1+i) ?

 

[loy]

help me ~~~

 

[HallsofIvy]

Yes, that is the "complex plane" where we represent the complex number z= x+iy by the point (x, y). So the number 1/2(1+ i)= 1/2+ (1/2)i is represented by the point (1/2, 1/2).

A circle with center at (a, b) and radius r has equation $(x- a)^2+ (y- b)^2= r^2$ so a circle with center at (1/2, 1/2) has equation $(x- 1/2)^2+ (y- 1/2)^2= r^2$. You may also know that the "modulus" or "absolute value" of z= x+ iy is $|z|= \sqrt{x^2+ y^2}$ so that circle can also be written as $|z- (1/2)(1+ i)|= r$.

 

[loy]

thanks very much
=)