Calculator Input
Formula Used
Rectangular form: z = a + bi
Modulus: r = √(a² + b²)
Argument: θ = atan2(b, a)
Polar form: z = r(cos θ + i sin θ) = r cis θ
De Moivre power rule: zⁿ = rⁿ(cos nθ + i sin nθ)
All m-th roots: wₖ = r^(1/m)[cos((θ + 2πk)/m) + i sin((θ + 2πk)/m)]
Root index: k = 0, 1, 2, ..., m - 1
How to Use This Calculator
- Enter the real part of the complex number in the first field.
- Enter the imaginary coefficient in the second field.
- Choose the integer power for De Moivre’s theorem.
- Enter the root degree to list all complex roots.
- Select radians or degrees for angle display.
- Choose the decimal places for rounded output.
- Press Calculate to show results above the form.
- Use the graph, CSV, and PDF buttons for review.
Example Data Table
| Real a | Imaginary b | Power n | Root degree m | Expected Use |
|---|---|---|---|---|
| 3 | 4 | 3 | 4 | Find z³ and four fourth roots. |
| 1 | 1 | 5 | 3 | Study angle multiplication and cube roots. |
| -2 | 2 | 4 | 6 | Check quadrant behavior and root spacing. |
| 0 | 5 | 2 | 5 | Analyze a pure imaginary number. |
De Moivre’s Theorem in Complex Number Work
Core Idea
De Moivre’s theorem links powers of complex numbers with trigonometry. It says that a complex number in polar form can be raised to an integer power by raising its modulus and multiplying its argument. This makes repeated multiplication faster and clearer. It also makes roots easier, because each root follows from dividing the adjusted angle by the root degree.
Why Polar Form Matters
A complex number a + bi has a distance from the origin. This distance is the modulus. It also has an angle from the positive real axis. This angle is the argument. Rectangular form is useful for addition. Polar form is better for powers, rotations, and roots. When the angle grows, the point rotates around the complex plane. When the modulus changes, the point moves closer or farther from the origin.
What This Calculator Helps You Check
This calculator converts the entered rectangular number into polar form. It then applies De Moivre’s theorem for a selected power. It also lists all roots for the chosen root degree. You can compare rectangular answers, polar answers, modulus changes, and angle changes. The graph gives a visual check. It plots the original number, the powered result, and the root points. This helps students see rotation, scaling, and symmetry.
Practical Math Uses
De Moivre’s theorem appears in algebra, trigonometry, engineering, signals, and physics. It is helpful when solving equations such as z^n = w. It also supports quick expansion of trigonometric identities. Complex roots often appear evenly spaced around a circle. That pattern is easier to understand when the answers are plotted. The exported CSV helps save numerical values. The PDF button helps create a clean study record.
Accuracy Notes
Angles can be displayed in degrees or radians. Small rounding differences may appear because trigonometric values use decimal approximations. Increase the decimal places for cleaner comparisons. For zero values, negative powers are not valid because division by zero would occur. Root calculations still need a positive degree. Always check whether your exponent, root degree, and rounding match the problem statement.
Use the table to compare values before copying final answers into assignments.
FAQs
1. What is De Moivre’s theorem?
It is a rule for raising complex numbers in polar form to integer powers. It raises the modulus to the power and multiplies the argument by that power.
2. Why does the calculator use polar form?
Polar form makes powers and roots easier. It separates distance from angle, so scaling and rotation can be handled with simple formulas.
3. Can I enter negative powers?
Yes, negative powers work for nonzero complex numbers. They represent reciprocal powers. A negative power is not valid when the complex number is zero.
4. What does root degree mean?
The root degree tells how many equal roots to calculate. For example, degree four gives four complex fourth roots, counted from k = 0 to k = 3.
5. Why are complex roots spaced around a circle?
Each root uses an angle separated by 2π divided by the root degree. This creates equal angular spacing around the same modulus circle.
6. Should I use degrees or radians?
Either display works. Radians are common in advanced math. Degrees are often easier for quick visual checks and classroom explanations.
7. Why do some results show tiny decimals?
Trigonometric calculations use decimal approximations. Very small values near zero may appear because of rounding. Increase or reduce decimal places as needed.
8. What does the graph show?
The graph plots the original number, the powered result, and all roots. It helps show rotation, scaling, quadrant location, and root symmetry.