How to use
- Type Z₁ and Z₂ in rectangular form a+bi, for example 3+4i, −2.5i, 5 or 3−4j. The imaginary unit can be i or j, in upper or lower case.
- You can also type polar form r∠θ, such as 5∠53.13. The angle is read in the selected angle unit; add ° or rad after it to choose the unit yourself (2∠30° or 2∠0.5rad).
- Pick an operation. The Z₂ box appears only for +, −, × and ÷, and the exponent n only for Z₁ⁿ. Functions such as the conjugate, square root, reciprocal, e^Z₁ and ln Z₁ use Z₁ alone.
- The result updates as you type: rectangular form a+bi first, then the polar form r∠θ, exponential form r·e^(iθ), modulus and argument, to 12 significant digits.
- The angle unit (degrees by default, or radians) applies to the argument, to the polar and exponential forms shown, and to any angle you type.
Formulas: rectangular form, polar form and De Moivre’s theorem
- Addition and subtraction work part by part: (a + bi) ± (c + di) = (a ± c) + (b ± d)i
- Multiplication expands the brackets and uses i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
- Division multiplies the top and bottom by the conjugate c − di: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
- Modulus |z| = r = √(a² + b²) and argument θ = atan2(b, a), given in the range −180° < θ ≤ 180°
- Polar and exponential form (Euler’s formula): z = r(cos θ + i sin θ) = r·e^(iθ)
- De Moivre’s formula zⁿ = rⁿ(cos nθ + i sin nθ). Also ln z = ln|z| + i·arg z and e^(a + bi) = e^a(cos b + i sin b).
Step-by-step example 1: multiplying and dividing complex numbers
Work through the numbers the calculator opens with, Z₁ = 3 + 4i and Z₂ = 1 − 2i.
- Expand the product: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i²
- Replace i² with −1: 3 − 2i + 8 = 11 − 2i. In polar form this is √125∠−10.3048464688° ≈ 11.1803398875∠−10.30°.
- To divide, multiply the top and bottom by the conjugate of the divisor, 1 + 2i. The top becomes (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i and the bottom (1 − 2i)(1 + 2i) = 1 + 4 = 5.
- So (3 + 4i) ÷ (1 − 2i) = (−5 + 10i) ÷ 5 = −1 + 2i.
- Addition and subtraction are simpler: (3 + 4i) + (1 − 2i) = 4 + 2i and (3 + 4i) − (1 − 2i) = 2 + 6i.
Step-by-step example 2: polar form and De Moivre’s formula
- Modulus of 3 + 4i: r = √(3² + 4²) = √25 = 5
- Argument: θ = atan2(4, 3) ≈ 53.1301023542° (0.927295218002 rad), so 3 + 4i = 5∠53.13° = 5·e^(i·53.13°).
- Going the other way, 2∠30° = 2(cos 30° + i sin 30°) = √3 + i ≈ 1.73205080757 + i.
- De Moivre: 1 + i = √2∠45°, so (1 + i)⁸ = (√2)⁸∠(8 × 45°) = 16∠360° = 16.
- In the same way (2∠30°)³ = 2³∠90° = 8i, and for square roots √(3 + 4i) = 2 + i and √(−4) = 2i.
Common mistakes and good to know
- Check the angle unit. In radian mode, 2∠30 means 30 radians, not 30°. If in doubt, type the unit: 2∠30°.
- The argument is not just arctan(b/a). For −3 − 4i, arctan(4/3) gives 53.13°, but the point is in the third quadrant, so arg = atan2(−4, −3) ≈ −126.869897646°.
- Rounded angles give rounded results: 5∠53.13 converts to 3.00000714566 + 3.99999464074i, not exactly 3 + 4i. Multiples of 90° are converted exactly (2∠90 = 2i).
- Write each part only once: 3+4i, 4i+3, 3+j4 and 2*i all work, but 3+4i+2 does not.
- π and fractions can’t be typed, so use decimals such as 3.14159265359 or 0.5. Tiny rounding leftovers are cleared, so e^(3.14159265359i) shows −1.
- Integer powers up to ±1024 are computed exactly by repeated multiplication, for example (3 + 4i)² = −7 + 24i.
Frequently asked questions
How do I convert a complex number to polar form?
Enter it as Z₁ and choose “Convert Z₁”: the calculator shows r∠θ, the exponential form, the modulus and the argument. For example, 3 + 4i has r = √(3² + 4²) = 5 and θ ≈ 53.13°, so it equals 5∠53.13°.
How do I convert polar form to rectangular form a+bi?
Type the number as r∠θ, for example 2∠30, and choose “Convert Z₁”; the main result is shown as a+bi. Because a = r cos θ and b = r sin θ, 2∠30° = √3 + i ≈ 1.73205080757 + i.
How do I raise a complex number to a power with De Moivre’s theorem?
Choose “Z₁ⁿ (power)” and enter the exponent n. By De Moivre’s formula zⁿ = rⁿ(cos nθ + i sin nθ), (1 + i)⁸ = (√2)⁸∠360° = 16; integer exponents are exact, and non-integer ones such as 0.5 give the principal value.
Can I use j instead of i?
Yes, 3−4j and 3−4i are treated as the same number. Electrical engineers usually write the imaginary unit as j so it isn’t confused with the symbol i for current.
Why is the argument of −3 − 4i −126.87° and not 53.13°?
Because the argument is found with atan2(b, a), which takes the quadrant into account. −3 − 4i lies in the third quadrant, so its argument is about −126.869897646°, and the calculator always gives arguments greater than −180° and at most 180°.
Which value is shown for square roots, logarithms and fractional powers?
The principal value, whose argument is greater than −180° and at most 180°. For example, √(−4) = 2i (the other root is −2i) and ln(−1) = πi ≈ 3.14159265359i.
Why do I get an error for division by zero or ln 0?
Because dividing by 0 + 0i and taking the logarithm of 0 are both undefined. No power of e equals 0, so ln 0 does not exist, and 0 has no direction, so its argument is undefined as well.