Maths / Linear algebra
Complex Number Calculator: Arithmetic, Polar Forms and Roots
Add, multiply and divide complex numbers, calculate powers and roots, find conjugates and arguments, and convert between rectangular and polar forms in degrees or radians.
Complex Number Calculator: Arithmetic, Polar Forms and Roots: A complex number has a real part and an imaginary part: 3 + 4i has real part 3 and imaginary part 4. The defining rule i² = −1 turns products into ordinary real arithmetic. For example, (1 + 2i)(3 − i) = 5 + 5i. The rectangular form is useful when adding signals or combining real and imaginary components. The polar form r∠θ instead describes the distance from zero and the direction in the complex plane, which is useful for powers, rotations and phasors. The starting example gives an answer immediately and can be edited. Choose the root order to list all n roots, whose angles are evenly spaced around a circle. Use the [scientific calculator](/scientific-calculator) for the other modes, including exact rational calculations and matrices. Runs 100% locally in your browser with zero server file uploads.
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More functions, constants and conversions
Use semicolons between arguments: ncr(52;5), dms(30;15;30), randint(1;6). Rnd rounds to the selected significant digits. E is Euler's number until you store a value in E; C is the speed of light until you store C.
- GCD
- 6
- LCM
- 144
- Integer quotient and remainder
- 2; 12
- 48 = 2 × 18 + 12
- 18 = 1 × 12 + 6
- 12 = 2 × 6 + 0
Constants: c, h, hbar (ħ), qe (elementary charge), me, mp, na, kb, gasr, grav (G), g (standard gravity), eps0, mu0, amu and sigma. Measured constants carry uncertainty; the displayed values are not exact. NIST / CODATA constants
Type or tap. Multiplication can be implied, as in 2π or 3(4+1); ^ is a power, ! a factorial, and % divides by 100. Enter calculates; Escape clears.
- Rectangular
- 5 + 5i
- Polar r∠θ
- 7.07106781187 ∠ 45
- Conjugate
- 5 − 5i
- All roots
- 2.45673236351 + 1.01761186409i; -2.45673236351 − 1.01761186409i
Complex arithmetic
OpenStax College Algebra, Complex Numbers (https://openstax.org/books/college-algebra-2e/pages/2-4-complex-numbers), explains rectangular arithmetic and conjugates. The text is openly licensed under CC BY 4.0.
Branches and elementary functions
NIST DLMF chapter 4 (https://dlmf.nist.gov/4) documents elementary functions and principal branches. Results here use decimal arithmetic; powers and roots can carry rounding error.
How to use it
- Type an expression using i, such as (1+2i)*(3-i).
- Choose DEG or RAD and enter a root order, or enter a polar radius and angle.
- Read the rectangular answer, magnitude, argument, conjugate and all roots.
Privacy & limitations
Expressions and results stay in your browser. Nothing is uploaded.
Related tools
Frequently asked questions
How do I enter imaginary numbers?
Use i after a number, such as 2i, or write 1+2i. Multiplication can be explicit or implied. Use parentheses around sums when multiplying them. sqrt(-4) returns 2i in Complex mode; ordinary Calculate mode rejects a real square root of a negative number.
What is the argument of a complex number?
The argument is the angle measured from the positive real axis. DEG displays degrees and RAD displays radians. The displayed principal angle lies between −180° and 180°, or between −π and π. The direction of zero is mathematically undefined.
Why does an nth root have several answers?
Adding a full turn does not change a complex number. Dividing each possible angle by n gives n different roots for a nonzero number. The single root function selects the principal branch; the All roots result lists every branch.
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