What is the Complex Number Plane?
Quick Answer: An interactive Argand diagram where complex numbers z = x+iy are plotted as points (x,y). It visualises vector addition (parallelogram law) and multiplication (scaling & rotation) in real time. Drag points or use sliders to explore how operations affect magnitude and angle.
Theory of Complex Numbers
A complex number z = x + iy is represented on the Argand plane with real part x on the horizontal axis and imaginary part y on the vertical axis. Its modulus is |z| = sqrt(x² + y²) and its argument (angle) is arg(z) = atan2(y, x). Addition of complex numbers follows vector parallelogram law: z1 + z2 = (x1+x2) + i(y1+y2). Multiplication in polar form: multiply moduli and add angles: |z1*z2| = |z1| |z2|, arg(z1*z2) = arg(z1) + arg(z2). This tool dynamically displays these relations.
Step-by-Step Examples
Example 1: Visualising Addition
- Set z1 = 2 + 1.5i, z2 = -1 + 2i. Observe the two vectors (red and blue).
- The yellow vector represents z1+z2 = 1 + 3.5i, completing a parallelogram.
- Drag z2's tip to see how the sum vector changes in real time.
Example 2: Multiplication = Rotation + Scaling
- Set z1 = 1 + i (magnitude √2, angle 45°), z2 = 0 + 1i (magnitude 1, angle 90°).
- The product z1*z2 = -1 + i (magnitude √2, angle 135°). Notice the angle addition and magnitude multiplication.
- Toggle product display to see the green vector for z1*z2.
Example 3: Conjugate and Reflection
- Observe that the conjugate (x - iy) of a point is simply reflected across the real axis. Though not explicitly drawn, you can mentally visualise by changing the sign of the imaginary part.
Frequently Asked Questions
What is the Argand diagram?
It's a plane where the x-axis represents the real part and the y-axis the imaginary part of a complex number, turning complex numbers into geometric points/vectors.
How does addition work visually?
Place the tail of the second vector at the head of the first; the sum is the vector from the origin to that new head. This forms a parallelogram.
Why does multiplication cause rotation?
In polar form, multiplying two numbers multiplies their distances from the origin and adds their angles. So the result is a rotated and scaled version.
Can I drag the points directly?
Yes, click and drag the red (z1) or blue (z2) point to interactively change their values. The sliders update accordingly.
What is the modulus and argument?
Modulus is the distance from origin; argument is the counterclockwise angle from the positive real axis. They are shown in the info overlay when you hover or drag a point.