Complex Numbers Circle Equation

In summary, the equation of a circle in complex number notation can be found by using the formula |z-a| = r, where a is the center point and r is the radius. In this case, with the given points (1,0), (0,1), and (0,0), the center is located at (1/2, 1/2) and the radius is 1/sqrt(2). Therefore, the equation of the circle is |z-(1/2)(1+i)| = 1/sqrt(2).
  • #1
jsi
24
0

Homework Statement



Write the equation of a circle in complex number notation: The circle through 1, i, and 0.

Homework Equations





The Attempt at a Solution



I know the equation for a circle with complex numbers is of the form |z-a| = r where a is the center point and r is the radius. I don't know how I'd go about finding it where they only give you 3 points like this. I assume the points correspond to complex numbers like w = x + iy == (x,y) so they'd be (1,0) (0,1) and (0,0)? I'm not sure where to go next and any help would be appreciated! thanks!
 
Physics news on Phys.org
  • #2
jsi said:

Homework Statement



Write the equation of a circle in complex number notation: The circle through 1, i, and 0.

Homework Equations





The Attempt at a Solution



I know the equation for a circle with complex numbers is of the form |z-a| = r where a is the center point and r is the radius. I don't know how I'd go about finding it where they only give you 3 points like this. I assume the points correspond to complex numbers like w = x + iy == (x,y) so they'd be (1,0) (0,1) and (0,0)? I'm not sure where to go next and any help would be appreciated! thanks!

Yes, think of the points (1,0), (0,1) and (0,0). Think of doing some geometry to find the center. E.g. the perpendicular bisector of each segment connecting two of those points goes through the center, right?
 
  • #3
ok, so then would the center be at (1/2, 1/2)?
 
  • #4
ok, I figured it out. Geometrically I found the center to be (1/2, 1/2) and then used that to find the radius from that point to a different point, (0,0), since that's easy to do then did sqrt((1/2)^2+(1/2)^2) to find the length of it which came out to 1/sqrt(2) so then the equation is just |z-(1/2)(1+i)| = 1/sqrt(2). Thanks for your help!
 

1. What is the equation of a complex number circle?

The equation of a complex number circle is z = a + bi, where a and b are real numbers and i is the imaginary unit.

2. How do you plot a complex number circle?

To plot a complex number circle, first determine the center of the circle, which is the point (a, b). Then, plot points on the circle by using the radius as the distance from the center and the angle as the direction from the positive real axis. This can be represented as r(cos θ + i sin θ), where r is the radius and θ is the angle in radians.

3. What is the relationship between the complex number circle and the Argand plane?

The complex number circle is a representation of complex numbers on the Argand plane, which is a coordinate system where the real numbers are shown on the x-axis and the imaginary numbers are shown on the y-axis. The complex number circle is centered at the origin of the Argand plane.

4. Can a complex number circle have a negative radius?

No, a complex number circle cannot have a negative radius. The radius of a complex number circle is always a positive real number, representing the distance from the center to any point on the circle.

5. What is the significance of the radius and center of a complex number circle?

The radius of a complex number circle represents the magnitude of the complex number, while the center represents the real and imaginary parts of the complex number. The distance from the center to any point on the circle is the same, representing the modulus or absolute value of the complex number.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
892
  • Calculus and Beyond Homework Help
Replies
27
Views
609
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
16
Views
3K
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
996
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Back
Top