Polar Form Calculator
Convert a + bi to r(cos theta + i sin theta) with modulus and angle steps.
Convert a + bi to r(cos theta + i sin theta) with modulus and angle steps.
- Grade level
- 9–12
- Curriculum band
- Difficulty
- Intermediate
- For this topic
- Time to master
- 2 weeks
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
Inputs — Polar Form Calculator
Result Summary
Results
Magnitude r
5
Polar radius
3 + 4i
Converted form
Closed-form complex arithmetic with quadrant-safe atan2 and zero guards; display rounds to six decimals.
Statistics
Live readouts from your current inputs.
Magnitude
5
r
Visualizations
Components vs magnitude
Magnitude size
Formula
Convert a + bi to r(cos theta + i sin theta) with modulus and angle steps.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| z | Complex number | a + bi | Point on the Argand plane |
| r | Modulus | number | Distance from origin |
| theta | Argument | radians | Angle from positive real axis |
Explanation
Rectangular form adds component-wise; polar form multiplies moduli and adds angles.
Derivation
i^2 = -1 turns FOIL products into (ac - bd) + (ad + bc)i; conjugates clear denominators; atan2 resolves quadrants.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Read the inputs into working variables.
Apply the defining formula for this operation.
Evaluate with real arithmetic throughout.
Cross-check with the polar view (modulus and angle).
Round for display to six decimals.
State the finished answer.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Run the page defaults once.
Inputs
Calculation
Apply the formula directly
Result
See live result
Interpretation
Confirms the engine path end to end.
Understanding Your Result
Your Result: Magnitude r: 5
Typical range: Modulus is non-negative; principal argument stays in (-pi, pi].
What This Means
The headline value is the exact result of the stated complex operation for your inputs.
Why It Matters
AC circuits, signal processing, control theory and quantum amplitudes all compute with complex arithmetic.
Professional Interpretation
Check the Argand vector: length is modulus, direction is argument.
Recommendations
- •Sketch the Argand vector before trusting angles
- •Use polar form for powers and roots
- •Rationalize division with the conjugate
Learn — The Complete Guide
Polar Form Calculator works on the Argand plane where the x-axis is real and the y-axis is imaginary.
The core concept
Every operation has a rectangular recipe and a polar meaning: shifts for addition, scaling plus rotation for multiplication.
- Argand plane maps a + bi to (a, b)
- Modulus is length; argument is direction
- Polar form makes powers trivial
Derivations — why the formulas work
Why FOIL gives (ac - bd)
- 1(a + bi)(c + di) = ac + adi + bci + bdi^2
- 2Replace i^2 with -1
- 3Group: (ac - bd) + (ad + bc)i
Where you meet this in real life
Learn this first — prerequisites in order
- 1Real arithmetic
- 2Pythagoras
- 3Sine and cosine
Careers that use this daily
- •Electrical engineer — phasors
- •Signal engineer — spectra
- •Physicist — wave functions
- •Control engineer — stability
Why it helps
- •Live Argand feedback
- •Polar cross-checks
- •Sign-safe quadrant logic
Best practices
- •Draw the vector first
- •Prefer polar form for powers
- •Guard zero denominators
Limitations — what this tool does not do
- •Principal argument only (-pi, pi]
- •Display rounds at six decimals
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Algebra 2 | 9–10 | HSN-CN | Rectangular ops · Modulus |
| Precalculus | 11–12 | HSN-CN | Polar and Euler · De Moivre |
Timeline — how this idea grew up
1545
Imaginary roots appear
Cardano meets square roots of negatives.
1748
Euler formula
e^(i theta) links exponentials and trig.
A little history
- • Argand (1806) gave complex numbers their plane.
- • Gauss made the geometric view standard.
Exam-day checklist
- • Give argument with range
- • Show conjugate rationalization
- • Check modulus after ops
Myths vs facts
Practical Tips
- •Sketch the vector before computing
- •Check modulus after every op
- •Use degrees for intuition, radians for trig
- •Conjugate to clear denominators
- •Copy results with the share bar
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
What is polar form?
Why use polar form?
How do I use this calculator?
Does it show steps?
Can I print or share results?
Is my data stored?
Related Calculators
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Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Modulus
- Distance from origin, |a + bi|.
- Argument
- Angle from positive real axis.
- Conjugate
- a - bi, the real-axis mirror.
- Polar form
- r(cos theta + i sin theta).
- Euler form
- r e^(i theta).
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
i rotates by 90 degrees
Multiplying by i turns the Argand vector a quarter-turn with no length change.
Euler ties five constants
e^(i pi) + 1 = 0 links e, i, pi, 1 and 0 in one line.
Modulus is Pythagoras
|a + bi| is the hypotenuse of legs a and b.
Compare Side by Side
Three views of the same answer — form, size and direction agree.
| View | Form | Value | Note |
|---|---|---|---|
| Rectangular | — | primary | a + bi |
| Modulus | 5 | length | |z| = r |
| Angle | 53.130102 deg | direction | theta |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
Polar Form Calculator computed
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Real arithmetic fluency
- ▸Pythagoras theorem
- ▸Basic trigonometry
Mastery progress — 0%
- I add and multiply without sign slips
- I convert rectangular <-> polar
- I apply De Moivre for powers
- 1
Week 1 · ~20 min/day
Rectangular ops
Goals
- Add and subtract component-wise
- Multiply with FOIL and i^2 = -1
- Conjugate and divide safely
Practice
Ten mixed ops daily on this page.
- 2
Week 2 · ~20 min/day
Polar power
Goals
- Convert both directions
- Apply De Moivre powers
- List all n-th roots
Practice
Convert each result back to check.
Exam tips
- •State principal argument range
- •Show conjugate step in division
Practice Problems
0/3 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Compute (3 + 2i) + (1 + 4i).
Compute (2 + 3i)(1 + 4i).
Write 3 + 4i in polar form.
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| |a + bi| = sqrt(a^2 + b^2) | Modulus | |3 + 4i| = 5 |
| conj = a - bi | Conjugate | conj(3 + 4i) = 3 - 4i |
| z = r(cos t + i sin t) | Polar | See live conversion |
| z^n = r^n(cos nt + i sin nt) | De Moivre | Powers in one line |
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Page Information
2026-09-29
3.1.0-math
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