What is my complex number answer?
Type the real and imaginary parts of two complex numbers and pick an operation. The complex number calculator gives the exact answer and its polar form.
- Result
- 23 − 14i
The result is 23 − 14i.
- Exact result
- 23 − 14i
- Real part x
- 23
- Imaginary part y
- -14
- Modulus |z|
- 26.92582404
- Argument θ (degrees)°
- -31.32869287
- Argument θ (radians)
- -0.5467888409
- Polar form
- 26.9258(cos −31.3287° + i sin −31.3287°)
- Conjugate
- 23 + 14i
Result: 23 − 14i. The result is 23 − 14i.
How to calculate
Adds, subtracts, multiplies and divides two complex numbers a + bi exactly, and gives the result’s modulus, argument, conjugate and polar form.
Example with the default inputs (Real part of z₁ (a) 4, Imaginary part of z₁ (b) 3, Operation z₁ × z₂, Real part of z₂ (c) 2, Imaginary part of z₂ (d) -5): The result is 23 − 14i.
Method: (a + bi) ± (c + di) = (a ± c) + (b ± d)i; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; (a + bi) ÷ (c + di) = ((ac + bd) + (bc − ad)i) ÷ (c² + d²).
- Each part is read as the exact decimal you typed, so +, −, × and ÷ have no rounding error.
- The argument is the principal value, from −180° to 180°.
- Each part is from −10¹² to 10¹².
Worked examples
Each example is checked against the calculator on every build.
- Real part of z₁ (a) 4, Imaginary part of z₁ (b) 3, Operation z₁ × z₂, Real part of z₂ (c) 2, Imaginary part of z₂ (d) -5 gives Result 23 − 14i, Real part x 23, Imaginary part y -14, Conjugate 23 + 14i, Modulus |z| 26.925824.Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02): (4 + 3i)(2 − 5i) = 23 − 14i
- Real part of z₁ (a) 2, Imaginary part of z₁ (b) 5, Operation z₁ ÷ z₂, Real part of z₂ (c) 4, Imaginary part of z₂ (d) -1 gives Exact result 3/17 + 22/17i, Result 0.1764705882 + 1.294117647i, Real part x 0.176471, Imaginary part y 1.294118.Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02): (2 + 5i) ÷ (4 − i) = 3/17 + 22i/17
- Real part of z₁ (a) 3, Imaginary part of z₁ (b) -4, Operation z₁ + z₂, Real part of z₂ (c) 2, Imaginary part of z₂ (d) 5 gives Result 5 + i, Real part x 5, Imaginary part y 1.Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02): (3 − 4i) + (2 + 5i) = 5 + i
- Real part of z₁ (a) 0.1, Imaginary part of z₁ (b) 0.2, Operation z₁ − z₂, Real part of z₂ (c) 0.3, Imaginary part of z₂ (d) 0.2 gives Result −0.2, Exact result −0.2, Real part x -0.2, Imaginary part y 0, Argument θ (degrees) 180.Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02)
- Real part of z₁ (a) 1, Imaginary part of z₁ (b) 1, Operation z₁ × z₂, Real part of z₂ (c) 1, Imaginary part of z₂ (d) 0 gives Modulus |z| 1.414214, Argument θ (degrees) 45, Polar form 1.41421(cos 45° + i sin 45°).Source: OpenStax, Algebra and Trigonometry 2e, §10.5 Polar Form of Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-5-polar-form-of-complex-numbers, retrieved 2026-10-02)
How it works
Type z₁ = a + bi and z₂ = c + di (each part from −10¹² to 10¹²) and pick an operation:
- Add: (a + c) + (b + d)i.
- Subtract: (a − c) + (b − d)i.
- Multiply: (ac − bd) + (ad + bc)i.
- Divide: ((ac + bd) + (bc − ad)i) ÷ (c² + d²). There is no answer when z₂ = 0 + 0i.
Each part is read as the exact decimal you typed, and the result x + yi is worked out in exact fractions, so there is no rounding error. Then:
- Result: x + yi with each part rounded half up (away from zero at a half) to 10 significant figures, from its exact value. An imaginary part of 1 is written i; a 0 part is left out; a negative part shows a minus sign (23 − 14i).
- Exact result: each part as its exact decimal when it ends within 15 significant figures, otherwise as a fraction in lowest terms (3/17 + 22/17i).
- Real part x and imaginary part y: rounded half up to 15 significant figures.
- Modulus |z| = √(x² + y²), to 10 significant figures.
- Argument θ: the angle of (x, y) from the positive real axis (atan2(y, x)), from −180° to 180°, in degrees and in radians, to 10 significant figures. For 0 it is 0.
- Polar form: r(cos θ + i sin θ) with r = |z| and θ in degrees, each to 6 significant figures, with the minus sign − (cos −31.3287°); just 0 when the result is 0.
- Conjugate: x − yi, written like the result.
There is no answer when a part of the result is larger than 10¹⁰⁰ (dividing by a number very close to 0).
Worked examples by hand
(4 + 3i) × (2 − 5i). x = 4 × 2 − 3 × (−5) = 8 + 15 = 23; y = 4 × (−5) + 3 × 2 = −20 + 6 = −14. Result 23 − 14i, conjugate 23 + 14i, modulus √(529 + 196) = √725 = 26.92582404.
(2 + 5i) ÷ (4 − i). c² + d² = 16 + 1 = 17. x = (2 × 4 + 5 × (−1)) ÷ 17 = 3/17; y = (5 × 4 − 2 × (−1)) ÷ 17 = 22/17. Result 0.1764705882 + 1.294117647i.
(3 − 4i) + (2 + 5i). (3 + 2) + (−4 + 5)i = 5 + i.
(0.1 + 0.2i) − (0.3 + 0.2i). (0.1 − 0.3) + (0.2 − 0.2)i = −0.2, exactly. It lies on the negative real axis, so θ = 180°.
(1 + i) × 1. The result is 1 + i, with modulus √2 = 1.414213562 and θ = 45°: polar form 1.41421(cos 45° + i sin 45°).
Other questions people ask
How do I multiply complex numbers?
Multiply out the brackets and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (4 + 3i)(2 − 5i): 8 − 20i + 6i − 15i² = 8 + 15 − 14i = 23 − 14i.
How do I divide complex numbers?
Multiply the top and the bottom by the conjugate of the bottom, which makes the bottom a real number. (2 + 5i) ÷ (4 − i) = (2 + 5i)(4 + i) ÷ (16 + 1) = (3 + 22i) ÷ 17 = 3/17 + 22/17 i.
How do I add or subtract complex numbers?
Add or subtract the real parts and the imaginary parts separately: (3 − 4i) + (2 + 5i) = (3 + 2) + (−4 + 5)i = 5 + i.
What is the modulus of a complex number?
Its distance from 0 in the complex plane: |x + yi| = √(x² + y²). For 23 − 14i it is √725 ≈ 26.9258.
What is the argument?
The angle θ from the positive real axis to the point (x, y), measured counterclockwise. The calculator gives the principal value, from −180° to 180°. For 1 + i it is 45°.
What is the conjugate?
The same number with the sign of the imaginary part flipped: the conjugate of x + yi is x − yi. A number times its conjugate is the real number x² + y².
Why can I not divide by 0 + 0i?
Division by z₂ means multiplying by 1 ÷ z₂ = (c − di) ÷ (c² + d²), and c² + d² is 0 only when z₂ is 0. Like dividing by the real number 0, there is no answer.