Set of Points in complex plane

[monnapomona]

Homework Statement

Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

Homework Equations

|z| = sqrt(x² + y²)
z = x + iy

The Attempt at a Solution

Tried to put absolute values on every thing by the Triangle inequality
|z| - |1| + |i| = |1|
sqrt(x² + y²) - 1 + 1 = 1
sqrt(x² + y²) = 1

Not sure if I'm approaching this question correctly...

 

[jbunniii]

The triangle inequality $|z + w| \leq |z| + |w|$ won't help you, because it's an inequality. The set of points with $|z - 1 + i| = 1$ is not the same as the set of points with $|z| + |1| + |i| = 1$.

A better approach is to recognize that
$$|z - 1 + i| = 1$$
if and only if
$$|z - 1 + i|^2 = 1$$
The squared equation is easier to work with, because for any complex number $w$ we have $|w|^2 = w\overline w$, where $\overline w$ is the complex conjugate of $w$.

 

[Ray Vickson]

Homework Statement

Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

Homework Equations

|z| = sqrt(x² + y²)
z = x + iy

The Attempt at a Solution

Tried to put absolute values on every thing by the Triangle inequality
|z| - |1| + |i| = |1|
sqrt(x² + y²) - 1 + 1 = 1
sqrt(x² + y²) = 1

Not sure if I'm approaching this question correctly...

No, you are not: $|z - 1 + i| = 1$ does NOT imply that $|z| -|1| + |i| = |1|$ or anything at all like it. In fact, it is almost always true that $|z_1 + z_2| \neq |z_1| + |z_2|$.

 

[PeroK]

Homework Statement

Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1

Not sure if I'm approaching this question correctly...

It says "describe" the set of points. Try thinking geometrically.

 

[Mark44]

It says "describe" the set of points. Try thinking geometrically.

And along these lines, an important use of the absolute value operation is to indicate the distance between two points. So |z - 1 + i| = 1 could also be written as |z - (1 - i)| = 1.