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How to use the complex number calculator
- Type an expression in the box. The answer updates as you type, in every form at once:
a + bi, polar in degrees and in radians, and exponential. - Pick Degrees or Radians. The switch sets the unit of any angle you type after ∠ and of the argument the tool reports.
- Open Working to see each operation with the numbers substituted. The Argand diagram beside it draws your inputs and the result.
- For a circuit, open the Phasors and AC circuits tab, set the frequency and the source voltage, then list the resistors, inductors and capacitors.
| You type | Meaning |
|---|---|
3 + 4i or 3 + j4 | Rectangular form; i and j both work |
5∠53.13 or 5<53.13 | Polar form, angle in the unit you picked |
5e^(0.9273i) | Exponential form, angle always in radians |
conj(z), abs(z), arg(z), re(z), im(z) | Conjugate, modulus, argument, real part, imaginary part |
z^n, sqrt(z), root(z, n) | Powers and roots; a root on its own lists all n roots |
exp(z), ln(z), cis(θ) | Exponential, principal natural log, cos θ + i sin θ |
Z1 || Z2 | Two impedances in parallel, Z₁Z₂/(Z₁ + Z₂) |
A product written without a sign is done first, as on a Casio-style calculator, so 1/2i means 1/(2i) = −0.5i. The "Read as" line under the answer shows how your input was grouped. If it is not what you meant, add brackets.
Rectangular, polar and exponential form
A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. Plot it as the point (a, b) on the Argand diagram: real part across, imaginary part up. The same point can also be given by its distance from the origin and its direction.
- Modulus: r = |z| = √(a² + b²), the length of the arrow.
- Argument: θ, the angle from the positive real axis, taken between −180° and 180°.
- Polar form: z = r∠θ = r(cos θ + i sin θ).
- Exponential form: z = r·e^(iθ), from Euler's formula e^(iθ) = cos θ + i sin θ, with θ in radians.
Each form suits a different job. Add and subtract in rectangular form, part by part. Multiply and divide in polar form: multiply the moduli and add the angles, or divide the moduli and subtract the angles. Division in rectangular form multiplies the top and the bottom by the conjugate of the bottom, a − bi, which turns the denominator into the real number a² + b². The working shows the rectangular method and then checks it in polar form.
Powers and roots use De Moivre's theorem: zⁿ = rⁿ∠(nθ). A nonzero number has exactly n different n-th roots, w_k = r^(1/n)∠((θ + 360°k)/n) for k = 0 to n − 1, evenly spaced on a circle of radius r^(1/n). The tool lists all of them and draws them. The root with k = 0 is the principal root, the one used inside a longer expression. More on the conversions in complex numbers in rectangular and polar form.
Phasors and impedance
In an AC circuit at angular frequency ω = 2πf, every voltage and current is a sine wave of the same frequency, so each one can be stored as a phasor: its size and phase as one complex number. Each part has an impedance: Z = R for a resistor, Z = jωL for an inductor and Z = 1/(jωC) = −j/(ωC) for a capacitor. Engineers write j because i already stands for current. Impedances in series add. In parallel their reciprocals, the admittances, add. Ohm's law becomes I = V/Z, and the complex power is S = V·I* = P + jQ with RMS values: P is real power in watts, Q is reactive power in var, |S| is apparent power in VA, and the power factor is P/|S|. The guide on complex arithmetic for engineering students has more circuit practice.
Three worked examples
1. Divide (2 + 3i) by (4 − 5i)
Multiply the top and the bottom by the conjugate of the bottom, 4 + 5i:
(2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i² = −7 + 22i(4 − 5i)(4 + 5i) = 4² + 5² = 41
So the quotient is −7/41 + (22/41)i = −0.1707317073 + 0.5365853659i. The polar check in the working agrees: 3.6055513∠56.309932° ÷ 6.4031242∠−51.340192° = 0.56309251∠107.65012°. The angle is past 90° because the real part is negative and the imaginary part positive, which puts the answer in the second quadrant.
2. All four roots of z⁴ = −16
In polar form −16 = 16∠180°. The fourth root of the modulus is 16^(1/4) = 2, and the angles are (180° + 360°k)/4 for k = 0, 1, 2 and 3: 45°, 135°, 225° and 315°. The tool returns:
- w₀ = 2∠45° = 1.414213562 + 1.414213562i, the principal root
- w₁ = 2∠135° = −1.414213562 + 1.414213562i
- w₂ = 2∠225° = −1.414213562 − 1.414213562i
- w₃ = 2∠315° = 1.414213562 − 1.414213562i
That is ±√2 ± √2i: the four corners of a square on a circle of radius 2. Raise any one of them to the fourth power and you get −16 back. The roots table prints 225° as −135° and 315° as −45° because it reports the principal argument.
3. A series RLC circuit on 230 V, 50 Hz
A 40 Ω resistor, a 200 mH inductor and a 100 µF capacitor are in series across a 230 V RMS supply at 50 Hz.
- ω = 2π × 50 = 314.15927 rad/s.
- ZL = j × 314.15927 × 0.2 = j62.83185 Ω and ZC = −j ÷ (314.15927 × 0.0001) = −j31.83099 Ω.
- Z = 40 + j(62.83185 − 31.83099) = 40 + j31.00086 Ω = 50.60685∠37.77646° Ω.
- I = 230∠0° ÷ 50.60685∠37.77646° = 4.544839∠−37.77646° A.
- S = V·I* = 826.22246 + j640.34026 VA, so P = 826.22 W, Q = 640.34 var and |S| = 1045.31 VA.
- Power factor = P ÷ |S| = 826.22246 ÷ 1045.31297 = 0.790407, lagging: the circuit is inductive, so the current lags the voltage.
Check: all the real power goes into the resistor, and I²R = 4.544839² × 40 = 826.22 W. The table under the result gives the voltage across each part: 181.79 V on R, 285.56 V on L and 144.67 V on C. These add up to far more than 230 V because they are out of phase; as phasors they sum to exactly 230∠0°. This pair of L and C would resonate at f₀ = 1/(2π√(LC)) = 35.588 Hz.
Reading the result and common mistakes
- The argument is not just arctan(b/a). For −1 − i, arctan((−1)/(−1)) = 45°, but the point is in the third quadrant, so θ = 45° − 180° = −135°. The working names the quadrant rule it used. See modulus and argument explained for every case, and converting a + bi to polar form for more practice.
- Degrees or radians. In degree mode, 5∠0.9273 is a point almost on the real axis. Switch to radians, or write the unit in the input:
5∠0.9273 rador5∠53.13°. An angle written with π, such as5∠pi/4, is always read in radians. - Exponential form uses radians.
5e^(0.9273i)is about 3 + 4i, while5e^(53.13i)is −4.809298179 + 1.36771745i. Type5e^(53.13°i)if your angle is in degrees. - Principal values. sqrt, ln, non-integer powers and a root inside a longer expression return the principal value.
(-8)^(1/3)gives 1 + 1.732050808i, not −2; the roots table lists −2 as the k = 1 root. - RMS or peak. Mains voltages such as 230 V and 120 V are RMS. A source written as v(t) = 10 cos(4t) gives a peak value, so choose Peak in the AC tab and the power is worked out with V/√2.
- Parallel binds tighter than + and −.
10 + j5 || -j10means 10 + (j5 ∥ −j10) = 10 + j10. Add brackets when in doubt.
Links that open a problem
The address bar updates as you work, and Copy link gives the same address, so a classmate or a student opens the exact problem. A plus sign in the link is read as a plus sign.
| Parameter | Example | Meaning |
|---|---|---|
z | ?z=(3+4i)*(1-2i) | The expression on the first tab |
ang | &ang=rad | Radians (degrees when left out) |
tab | ?tab=ac | Open the AC circuit tab |
f, fu | f=4&fu=rad/s | Frequency, in Hz unless fu=rad/s |
v, va, peak | v=10&va=0&peak=1 | Source size, phase in degrees, peak instead of RMS |
conn, p | conn=parallel&p=R20,L50mH | Series or parallel, and the parts: R, L, C with units, or Z(3+4j) |
Embed the Complex Number Calculator on your website
Teachers, bloggers and course sites can add this tool to any web page for free. Paste the code below into an HTML or embed block (WordPress, Google Sites, Canvas, Moodle, Blogger, Wix). It loads the same tool you see above, without our menus or ads. Please keep the credit line under the frame. Sizes, a live preview and step-by-step guides for each platform are on the calculator widget page.
Frequently asked questions
How do I convert a + bi to polar form?
Type the number, for example -1 - i. The polar card shows 1.414213562∠−135°, and the working shows the modulus √((−1)² + (−1)²) = √2 and the quadrant rule that gives −135° instead of 45°.
Can I enter phasors with j and an angle symbol?
Yes. Use j or i, and ∠ or the less-than sign for the angle, as in 10∠30 + 5∠-45 or 3 + j4. The keys under the box insert ∠, °, π and j on a phone. When you type j, the answers are written in j form too.
How do I find all the cube roots of a complex number?
Type the root on its own, for example root(8i, 3). The tool lists √3 + i, −√3 + i and −2i with the De Moivre working, and draws all three on a circle of radius 2.
Why does arctan(b/a) give the wrong argument?
The arctangent only returns angles between −90° and 90°, so it cannot tell −1 − i from 1 + i. The calculator checks the quadrant: it adds 180° in the second quadrant and subtracts 180° in the third, which gives the principal argument between −180° and 180°.
Does the AC tab use RMS or peak values?
RMS by default, which is how mains voltages are quoted. Choose Peak for a source given as an amplitude, such as 10 cos(4t). The currents are then peak values too, and the power uses half of V times the conjugate of I.
What is i to the power i?
A real number: i^i = e^(−π/2) = 0.2078795764 for the principal value. Type i^i and the working shows ln i = (π/2)i, then i·ln i = −π/2.