Subjects complex numbers

Complex Inversion 2D72E9

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Search Solutions

Complex Inversion 2D72E9


1. **Problem Statement:** Given a complex number $z_0$ on the positive real axis between 0 and 1, and points $A$, $B$, $C$, and $D$ representing complex numbers in the plane, determine which point corresponds to $\frac{1}{z_0}$. 2. **Recall the property of complex inversion:** For a complex number $z_0 = x$ (since it lies on the real axis, $x > 0$ and $0 < x < 1$), its reciprocal is $\frac{1}{z_0} = \frac{1}{x}$. 3. **Interpretation on the complex plane:** Since $z_0$ is on the positive real axis between 0 and 1, $x$ is a positive number less than 1. The reciprocal $\frac{1}{x}$ is therefore a positive real number greater than 1. 4. **Locate $\frac{1}{z_0}$:** On the real axis, $\frac{1}{z_0}$ lies to the right of 1, further along the positive real axis. 5. **Identify the point:** Among the points given, $A$ lies in the first quadrant (positive real and positive imaginary), $B$ is near the origin on the positive real side but close to the imaginary axis, $C$ is near the origin on the negative real side, and $D$ is in the second quadrant (negative real, positive imaginary). Since $\frac{1}{z_0}$ is a positive real number greater than 1, it must lie on the positive real axis to the right of $z_0$. The only point that fits this description is $A$. **Final answer:** Point $A$ represents $\frac{1}{z_0}$.