Complex Numbers Calculator
Calculate with complex numbers: type one expression — addition, subtraction, multiplication, division, real part, imaginary part, absolute value (magnitude), argument angle in radians or the conjugate — and read the answer in rectangular form a+bi plus polar form r∠φ° — all computed in this browser tab.
Example: (3+2i)+(5-3i), re(3+2i), abs(3+2i), arg(3+2i), conj(3+2i), i^2. Press Enter or = to calculate.
Latest result appears here.
Complex numbers calculation examples: addition (3+2i)+(5-3i), subtraction (3+2i)-(5-3i), multiplication (3+2i)*(5-3i), division (3+2i)/(5-3i), real part re(3+2i), imaginary part im(3+2i), absolute value abs(3+2i), argument angle in radians arg(3+2i), and conjugate conj(3+2i). Each result is shown in rectangular form a+bi plus polar form r∠φ° when it carries an imaginary part. All math happens in this browser tab — nothing is uploaded, stored, or shared.
Type one expression
Write a calculation like (3+2i)+(5-3i) or press the keypad keys (i, +, −, ×, ÷, xʸ, √, π, Re(, Im(, |z|, arg(, conj() to build it. The imaginary unit i satisfies i² = −1, and 2i is short for 2×i.
Press = or Enter
The result is appended to the calculation results tape below the pad. Real results show as a plain number; complex results also get the polar form r∠φ° with the angle in degrees.
Keep calculating
Each calculation adds a new block to the tape, so you can compare a whole session of work. Invalid input, division by zero or results beyond displayable size show a clear error and never touch the tape.
Complex numbers calculation examples
| Operation | Expression | Result |
|---|---|---|
| Addition | (3+2i)+(5-3i) | 8 - i |
| Subtraction | (3+2i)-(5-3i) | -2 + 5i |
| Multiplication | (3+2i)*(5-3i) | 21 + i |
| Division | (3+2i)/(5-3i) | 0.264705882353 + 0.558823529412i |
| Real part | re(3+2i) | 3 |
| Imaginary part | im(3+2i) | 2 |
| Absolute value (magnitude) | abs(3+2i) | 3.60555127546 |
| Argument angle (radians) | arg(3+2i) | 0.588002603548 |
| Conjugate number | conj(3+2i) | 3 - 2i |
Complex number forms
A complex number is written z = a + bi, where a is the real part, b is the imaginary part and i = √−1. Its absolute value (magnitude) is |z| = √(a² + b²) and its argument angle is arg(z) = atan2(b, a) in radians. The same number in polar form is z = r∠φ with radius r = |z| and angle φ in degrees — e.g. 3+2i has magnitude √13 ≈ 3.60555127546 and angle atan(2/3) ≈ 0.588002603548 rad ≈ 33.69°. Division multiplies by the conjugate: (a+bi)/(c+di) = (a+bi)(c−di)/(c² + d²).
How this calculator works
- Which functions can I use?
- re(z) takes the real part, im(z) the imaginary part, abs(z) the magnitude, arg(z) the angle in radians, conj(z) the conjugate a−bi, and sqrt(z) the principal square root — sqrt(-4) = 2i. Constants: i (imaginary unit) and pi (π); the xʸ key inserts a power, e.g. i^2 = -1.
- Why does every complex result show a second line with ∠?
- That is the polar form r∠φ°: the magnitude and the angle in degrees, with the angle in (−180°, 180°]. It is shown only when the result actually has an imaginary part, so re(3+2i) = 3 stays a plain number.
- Is arg(z) in degrees or radians?
- The arg(z) function returns radians (like the reference tool); the ∠ line of each result shows the same angle converted to degrees. arg(-4) = π ≈ 3.14159265359 rad = 180°.
- How precise are the results?
- Numbers display up to 12 significant digits, switching to exponential form outside 1e-6…1e12, so (3+2i)/(5-3i) shows the exact-looking 0.264705882353 + 0.558823529412i (= 9/34 + 19i/34). Division by zero and un-displayably huge results are refused with a clear error instead of printing Infinity.
Privacy: this calculator runs entirely in your browser tab. Your input and results are never uploaded, stored, or shared.