Complex Number Calculator — Add, Multiply, Modulus & Polar
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A complex number calculator works with numbers of the form a + bi, where i² = −1. To multiply (2 + 3i)(4 − i), expand: 8 − 2i + 12i − 3i². Since −3i² = +3, the product is 11 + 10i. Its modulus, the distance from 0, is √(11² + 10²) = √221 ≈ 14.87.
(2 + 3i) × (4 − i) = 11 + 10i.
Show steps
- Multiply out (FOIL): (2 + 3i)(4 − i) = 8 − 2i + 12i − 3i².
- Replace i² with −1: 3i² becomes 3.
- Collect real and imaginary parts: (8 + 3) + (−2 + 12)i = 11 + 10i.
- Modulus of the answer: |11 + 10i| = √(11² + 10²) = √221 ≈ 14.8661.
- Check with moduli: |A| × |B| = √13 × √17 = √221, which matches |A × B|. Multiplying multiplies the lengths and adds the angles.
How to use the complex number calculator
- Choose a tab: + − × ÷ for two numbers, One number for the conjugate, modulus, argument and polar forms, Powers & roots for zⁿ and all n roots, Expression to type something like (2 + 3i)(4 − i)/(1 + i), or Polar → a + bi to convert r and θ.
- Enter the real and imaginary parts. For 5 − 2i, type 5 and −2. Fractions such as 3/4 and decimals work, and answers stay exact.
- Pick degrees or radians at the top for angles. Read the answer in standard form, the decimal form, modulus, argument and polar form, look at the Argand diagram, and open Show steps.
Complex number rules at a glance
A complex number z = a + bi has a real part a and an imaginary part b. The key rule is i² = −1. Everything else follows from ordinary algebra.
| Operation | Rule | Example |
|---|---|---|
| Add | (a + bi) + (c + di) = (a + c) + (b + d)i | (3 − 5i) + (−1 + 2i) = 2 − 3i |
| Multiply | (a + bi)(c + di) = (ac − bd) + (ad + bc)i | (2 + 3i)(4 − i) = 11 + 10i |
| Conjugate | a + bi → a − bi | 3 + 4i → 3 − 4i |
| Modulus | |z| = √(a² + b²) | |3 + 4i| = 5 |
| Divide | multiply top and bottom by the conjugate of the bottom | (4 + 2i) ÷ (1 − i) = 1 + 3i |
| Powers of i | repeat every 4: i, −1, −i, 1 | i²³ = i³ = −i |
Multiplying complex numbers
Multiply each part of the first number by each part of the second (FOIL), then replace i² with −1 and collect terms. For (2 + 3i)(4 − i): 2 × 4 = 8, 2 × (−i) = −2i, 3i × 4 = 12i and 3i × (−i) = −3i². So the product is 8 + 10i − 3i² = 8 + 10i + 3 = 11 + 10i. A number times its conjugate is always real: (3 + 2i)(3 − 2i) = 9 + 4 = 13.
Dividing complex numbers
You cannot leave i in a denominator, so multiply the top and bottom by the conjugate of the bottom. For (4 + 2i) ÷ (1 − i), use 1 + i. The top becomes (4 + 2i)(1 + i) = 4 + 4i + 2i + 2i² = 2 + 6i, and the bottom becomes 1² + 1² = 2. Divide each part by 2: the answer is 1 + 3i. Check by multiplying back: (1 + 3i)(1 − i) = 1 − i + 3i − 3i² = 4 + 2i.
Modulus, argument and the Argand diagram
On an Argand diagram the real part goes across and the imaginary part goes up, so 5 − 2i is the point (5, −2). The modulus |z| is the distance from 0, found with the Pythagorean theorem: |3 + 4i| = √(9 + 16) = 5. The argument θ is the angle from the positive real axis.
Find the argument with arctan(|b| ÷ |a|), then adjust for the quadrant. In Quadrant I keep it, in Quadrant II subtract it from 180°, in Quadrant III subtract 180° from it, and in Quadrant IV make it negative. So arg(1 + i) = 45° (π/4), but arg(−1 + i) = 180° − 45° = 135° (3π/4), not −45° as a plain arctan(b/a) would suggest. The principal argument is between −180° and 180°.
Polar and exponential form
Using the modulus r and argument θ, z = r(cos θ + i sin θ). By Euler’s formula, e^(iθ) = cos θ + i sin θ, so the same number is z = re^(iθ). Engineers often write r∠θ. To convert back, a = r cos θ and b = r sin θ: for r = 2 and θ = 60°, a = 2 × 1/2 = 1 and b = 2 × √3/2 = √3, so z = 1 + √3 i. Polar form makes multiplication easy: multiply the moduli and add the angles.
Powers and De Moivre’s theorem
De Moivre’s theorem says zⁿ = rⁿ(cos nθ + i sin nθ). For (1 + i)⁸, r = √2 and θ = 45°, so r⁸ = 16 and 8θ = 360°, which points along the positive real axis: (1 + i)⁸ = 16. Powers of i cycle every 4, so divide the exponent by 4 and use the remainder: 23 = 4 × 5 + 3, so i²³ = i³ = −i.
Roots of complex numbers
Every nonzero complex number has exactly n different nth roots. They share the modulus r^(1/n) and sit evenly around a circle, 360°/n apart, at angles (θ + 360°k)/n. The cube roots of 8 are 2, −1 + √3 i and −1 − √3 i. The square roots of 3 + 4i are 2 + i and −2 − i, because (2 + i)² = 4 + 4i + i² = 3 + 4i. The equation x² + 1 = 0 has the two roots i and −i.
Common mistakes to avoid
- Forgetting that i² = −1. In (2 + 3i)(4 − i) the term −3i² is +3, not −3.
- Dividing part by part. (4 + 2i) ÷ (1 − i) is not 4 − 2i. Use the conjugate.
- Using arctan(b/a) without checking the quadrant. It gives the wrong angle for points left of the imaginary axis.
- Calling the imaginary part bi. The imaginary part of 3 + 4i is 4, not 4i.
- Stopping at one root. A cube root question has three answers.
Where complex numbers are used
Electrical engineers describe alternating current with complex numbers, usually writing j for i so it is not confused with current. A resistance and a reactance combine into one complex impedance, and its modulus and argument give the size and phase shift. Complex numbers also appear in signal processing, control systems and in solving any quadratic equation with a negative discriminant.
Complex numbers practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
FOIL: 3 − i + 6i − 2i² = 3 + 5i + 2 = 5 + 5i. Forgetting that i² = −1 gives 1 + 5i.
Multiply top and bottom by 1 − i: (6 + 2i)(1 − i) = 6 − 6i + 2i − 2i² = 8 − 4i, and the bottom is 2. So the answer is 4 − 2i.
The reference angle is 45° and the point is in Quadrant III, so θ = 45° − 180° = −135° (or 225°). A plain arctan(1) gives 45°, which is wrong.
r = √2 and θ = 45°, so r⁴ = 4 and 4θ = 180°: 4(cos 180° + i sin 180°) = −4.
Every nonzero complex number has exactly n distinct nth roots, so it has 3 cube roots, spaced 120° apart.
Quiz complete
Frequently asked questions
What is a complex number?
A number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. The real part is a and the imaginary part is b.
What are examples of complex numbers?
3 + 4i, 5 − 2i, −i and 1/2 + (3/4)i are all complex numbers. So are real numbers such as 7 (which is 7 + 0i) and pure imaginary numbers such as 2i (which is 0 + 2i).
How do you multiply complex numbers?
Expand with FOIL, replace i² with −1, and collect the real and imaginary parts: (2 + 3i)(4 − i) = 8 − 2i + 12i − 3i² = 11 + 10i.
How do you divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom so the denominator becomes real: (4 + 2i)/(1 − i) = (4 + 2i)(1 + i)/2 = (2 + 6i)/2 = 1 + 3i.
How do you find the argument of a complex number?
Plot the point, find the reference angle arctan(|b|/|a|), and adjust for the quadrant. arg(1 + i) = 45°, while arg(−1 + i) = 135° because the point is in the second quadrant.
Is 2 a complex number?
Yes. Every real number is a complex number with imaginary part 0, so 2 = 2 + 0i. Its modulus is 2 and its argument is 0.
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