VisuApps
Geometric visualisations of complex numbers
Apps
Proof of Euler's formula in eight steps: the Taylor series of eˣ only needs plus and times and therefore also works for complex numbers. With z = iφ every term turns another 90°, and the arrows form a spiral towards the unit circle. Sorted into terms without and with i, they become the series of cosine and sine, shown on a helix with cos and sin shadows. De Moivre's theorem follows at the end.
Complex numbersTry it
Add two complex numbers and understand the sum as the parallelogram rule.
Complex numbersTry it
Divide two complex numbers — magnitudes divide, angles subtract.
Complex numbersTry it
Multiply two complex numbers — magnitudes multiply, angles add.
Complex numbersTry it
Every representation and operation of complex numbers in one place: drag z and w in the complex plane and read or type the Cartesian, polar and Euler forms live. The angle trap shows where arctan(b/a) lands in the wrong quadrant. Addition shifts, multiplication rotates and stretches the whole plane, division rotates back, conjugation mirrors, powers form a spiral and roots a regular n-gon. Each result shows the Cartesian and the polar calculation side by side and marks the easier one. The ‘One number’ mode shows the building blocks of the conversion for a single z (legs a and b, pointer r, angle φ, unit circle) and converts step by step between Cartesian, polar and Euler form.
Complex numbersTry it
Visualise all n-th roots of a complex number as a regular polygon.
Complex numbersTry it
Raise a complex number to a power and explore the spiral of intermediate values.
Complex numbersTry it
The number systems ℕ, ℤ, ℚ, ℝ and ℂ as a ladder: above a 3D number line stands the modulus of the equation, and a missing solution stays a hollow ring until the next system supplies it (5 + x = 2, 3·x = 5, x² = 2, x² = −2). A zoom closes in on √2; in ℂ the line unfolds into a plane, and the surface |z² + 2| touches the floor at ±i√2. In the end stage, rolled polynomials show the fundamental theorem of algebra: as many roots as the degree.
Complex numbersTry itWant to use these apps in your courses? Let’s find out what works for you.
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