Complex Number Calculator
Enter numbers in the form a + bi to add, subtract, multiply, divide, raise to a power, find the conjugate, convert to polar form or compute n-th roots, with the working shown.
What is a complex number?
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. The number a is the real part and b is the imaginary part. Complex numbers can be drawn as points (a, b) in a plane, the Argand diagram, and they make it possible to solve every polynomial equation: the equation x² + 1 = 0 has the solutions i and −i.
Arithmetic with complex numbers
Addition and subtraction work on the real and imaginary parts separately. Multiplication uses the distributive law and i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division multiplies the numerator and denominator by the conjugate of the denominator, which turns the denominator into the real number c² + d². The calculator shows all of these substitutions with your numbers. Inputs may be integers, decimals or fractions, such as 1/2 − 3/4i, and all of these operations and integer powers are exact, so no rounding error appears.
Modulus, argument and polar form
The modulus |z| = √(a² + b²) is the distance from the origin, and the argument θ is the angle from the positive real axis. Then z = r(cos θ + i sin θ) = reiθ, which follows from Euler's formula. Polar form makes multiplication easy: moduli multiply and angles add. The modulus is displayed exactly, for example 3 + 4i has modulus 5 and 1 + i has modulus √2, while the argument is a decimal approximation, with the exact value in terms of π when it is a standard angle such as π/4.
Powers and roots with De Moivre's theorem
De Moivre's theorem states that (cos θ + i sin θ)n = cos nθ + i sin nθ. It gives integer powers directly and also shows that every non-zero number has exactly n complex n-th roots, equally spaced on a circle. For example the three cube roots of 8 are 2 and −1 ± √3 i. The calculator computes powers exactly and roots numerically, checking each root by raising it back to the n-th power.
Worked examples
These examples are computed by the same engine as the calculator above, so you can compare each step with your own working.
(3+4i)(1−2i)
- Operation:
z₁ × z₂ - z₁:
3+4i - z₂ (for + − × ÷):
1-2i - n (power or root index):
3
Step-by-step solution
Formula used
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
CalculateExpand with i² = −1
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
(3 + 4i)(1 − 2i)
real part: ac − bd = 3·1 − 4·−2 = 11
imaginary part: ad + bc = 3·−2 + 4·1 = −2
ResultResult
z₁ × z₂ = 11 − 2i
CheckCheck with moduli
|z₁|²·|z₂|² = 125 and |z₁z₂|² = 125 ✓ (the modulus of a product is the product of the moduli)
(1+2i) ÷ (3−i)
- Operation:
z₁ ÷ z₂ - z₁:
1+2i - z₂ (for + − × ÷):
3-i - n (power or root index):
3
Step-by-step solution
Formula used
z₁ / z₂ = z₁·conj(z₂) / |z₂|²
CalculateMultiply top and bottom by the conjugate of z₂
z₁/z₂ = z₁·z̄₂ / |z₂|²
conjugate of z₂ = 3 + i
numerator: (1 + 2i)(3 + i) = 1 + 7i
denominator: (3 − i)(3 + i) = 3² + −1² = 10
CalculateDivide each part by the real denominator
(1 + 7i) / 10 = 1/10 + (7/10)i
CheckVerify by multiplying back
(1/10 + (7/10)i) × (3 − i) = 1 + 2i = z₁ ✓
(1+i)^8
- Operation:
z₁ to the power n - z₁:
1+i - z₂ (for + − × ÷):
0 - n (power or root index):
8
Step-by-step solution
Formula used
zⁿ by exact multiplication; De Moivre: (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ)
CalculateExact repeated squaring
The power is computed exactly with square-and-multiply on the real and imaginary parts.
CheckCheck with De Moivre's theorem
zⁿ = rⁿ(cos nθ + i sin nθ): r = 1.4142136, θ = 0.78539816 gives about 16 + 0i, matching the exact result.
ResultResult
z^8 = 16
Polar form of −1+i
- Operation:
Modulus, argument, polar form of z₁ - z₁:
-1+i - z₂ (for + − × ÷):
0 - n (power or root index):
3
Argument θ
3π/4 ≈ 2.3561945 rad
135° (approximate)
Modulus is exact when it is a radical; the argument is a floating-point value, shown to 8 significant digits.
Step-by-step solution
Formula used
|z| = √(a² + b²), θ = atan2(b, a)
z = r(cos θ + i sin θ) = r·e^(iθ)
CalculateModulus
|z| = √(a² + b²)
|z| = √(−1² + 1²) = √2 = √2 ≈ 1.4142136
CalculateArgument (angle)
θ = atan2(b, a)
arg(z) = atan2(1, −1) ≈ 2.3561945 rad (135°) = 3π/4 [principal value in (−π, π]]
ResultPolar and exponential form
z = r(cos θ + i sin θ) ≈ 1.4142136·(cos 2.3561945 + i sin 2.3561945) = 1.4142136·e^(i·2.3561945)
CheckConvert back to a + bi
r cos θ = −1, r sin θ = 1 ✓
Cube roots of 8
- Operation:
n-th roots of z₁ - z₁:
8 - z₂ (for + − × ÷):
0 - n (power or root index):
3
Root k = 1
−1 + 1.7320508i
Root k = 2
−1 − 1.7320508i
Roots are floating-point approximations (8 significant digits). Sums, products, quotients and integer powers on this page are exact.
Step-by-step solution
Formula used
The n-th roots of z = r·e^(iθ) are r^(1/n)·e^(i(θ + 2πk)/n), k = 0, 1, …, n − 1
CalculateWrite z in polar form
z = r(cos θ + i sin θ)
r = |z| = 8
θ = arg z = 0 rad
CalculateApply the root formula
z^(1/n): rₖ = r^(1/n)·(cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)), k = 0, …, n − 1
modulus of each root: 8^(1/3) = 2
ResultThe 3 roots (approximate)
k = 0: 2 + 0i
k = 1: −1 + 1.7320508i
k = 2: −1 − 1.7320508i
CheckRaise each root to the power 3
Every root rₖ satisfies rₖ^3 ≈ z (largest relative error 1.5e-15).
The roots lie on a circle of radius 2, evenly spaced by 360°/3 = 120°.
Frequently asked questions
How do I enter a complex number?
Type it as 3+4i, 3 - 4i, -2i, i, 5 or 1/2 + 3/4i. The letter j is accepted as well. Use one real term and one imaginary term.
What is the conjugate of a complex number?
The conjugate of a + bi is a − bi. Multiplying a number by its conjugate gives the real number a² + b², the square of its modulus.
How do I divide complex numbers?
Multiply the numerator and the denominator by the conjugate of the denominator, then divide the real and imaginary parts by the real result c² + d².
Why are there n different n-th roots?
Because adding a full turn 2π to the argument gives the same number but a different angle after dividing by n. The n distinct results are spaced 360°/n apart.