Analyze i cycles, rotations, roots, and complex coordinates. See clear steps, tables, exports, and plots. Make imaginary number work faster, easier, and more accurate.
Choose a mode, enter values, and calculate powers, rotations, polar form, or n-th roots involving the imaginary unit.
| Mode | Example Input | Example Output |
|---|---|---|
| Power of i | n = 23, coefficient = 1 | i^23 = -i |
| Rotate z by i^n | z = 3 - 2i, n = 3 | (3 - 2i)i^3 = -2 - 3i |
| Complex to polar form | z = 4 + 4i | r = 5.656854, θ = 45° |
| n-th roots of ±i | n = 3, target = i | Roots lie on the unit circle at equal angles |
1) Powers of the imaginary unit
The imaginary unit repeats every four powers:
i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, and i^(n+4) = i^n.
2) Reduce the exponent with modular arithmetic
Compute the remainder:
r = n mod 4
Then use the cycle above to identify i^n.
3) Multiply a complex number by i^n
For z = a + bi:
z × i rotates 90° counterclockwise,
z × i^2 rotates 180°,
z × i^3 rotates 270° counterclockwise.
4) Convert cartesian to polar form
If z = a + bi:
r = √(a² + b²)
θ = atan2(b, a)
Polar form becomes:
z = r(cos θ + i sin θ) = re^(iθ)
5) n-th roots of i or -i
Using De Moivre’s theorem:
z_k = cis((θ + 2πk) / n), for k = 0, 1, ..., n-1
Use θ = π/2 for i and θ = -π/2 for -i.
1. Select the calculation mode that matches your problem.
2. Enter the exponent, complex number parts, or root index.
3. Click Calculate to generate the result instantly.
4. Review the output cards shown above the form.
5. Study the Plotly graph to understand the position or rotation visually.
6. Export the computed result using the CSV or PDF buttons.
7. Use the example table to compare your inputs with sample cases.
Because multiplying by i cycles through four values: 1, i, -1, and -i. After four multiplications, the sequence returns to the starting value and repeats.
Divide the exponent by 4 and use the remainder. A remainder of 0 gives 1, 1 gives i, 2 gives -1, and 3 gives -i.
It rotates the point 90 degrees counterclockwise on the complex plane. The number’s magnitude stays the same, but its direction changes.
Polar form makes multiplication, division, powers, and roots easier. It expresses a complex number using magnitude and angle instead of separate real and imaginary parts.
They are n equally spaced points on the unit circle. Each root has magnitude 1, and the angles differ by 360° divided by n.
Yes. The modular cycle still works for negative exponents after adjustment. The tool reduces the exponent correctly before mapping it to the repeating pattern.
No, multiplying by powers of i only rotates the number. Since |i| = 1, the magnitude remains unchanged while the angle shifts.
The graph places results on the complex plane. Depending on the mode, it shows the final point, both original and rotated points, or the full set of roots.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.