Complex Number Calculator
Operations on complex numbers
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Complex Number Calculator is built for operations on complex numbers — fast, free, and private. You supply Real 1, Imag 1 and Real 2 to Complex Number; the answer follows at once. Every Complex Number answer can be reproduced by hand. Students and professionals reach for Complex Number when speed matters. Nothing is uploaded while Complex Number works, and the figures stay put. Complex Number handles complex number calculator the same way as free online complex number calculator — one of the Math tools on CalcProMaster. Complex Number shows formula, assumptions and result together. Open Complex Number and enter your numbers.
What the defaults give
(3 + 2i) + (1 + 4i) = 4 + 6i.
The real parts add to 4 and the imaginary parts add to 6. Addition is done component by component, exactly like vectors.
The four operations
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
Multiplication: multiply out and use i² = −1
Division: multiply top and bottom by the conjugate of the denominator
Worked examples with the defaults
(3 + 2i) + (1 + 4i) = 4 + 6i
(3 + 2i) − (1 + 4i) = 2 − 2i
(3 + 2i) × (1 + 4i) = 3 + 12i + 2i + 8i² = −5 + 14i
(3 + 2i) ÷ (1 + 4i) = (3 + 2i)(1 − 4i) ÷ 17 = (11 − 10i) ÷ 17 ≈ 0.647 − 0.588i
Why i² = −1 does all the work
Complex numbers exist because x² = −1 has no real solution. Once i is defined as the number whose square is −1, every polynomial has a root and multiplication follows the ordinary distributive rule with that one substitution. The pattern in the example above — the imaginary terms collapsing into a real one — is the mechanism behind rotation, oscillation and alternating current.
Where complex arithmetic shows up
- Electrical engineering. Impedance combines resistance and reactance as a complex number, and division gives the phase shift.
- Signal processing. The Fourier transform maps a real signal onto complex frequencies.
- Geometry. Multiplying by i rotates a point 90° about the origin, which is why complex numbers describe rotation so cleanly.
- Control theory. Stability is read from the position of complex poles in the s-plane.
The conjugate — the same number with the imaginary sign flipped — appears in division, in modulus, and in finding real quadratic factors.
Addition is component-wise, so (3+2i) + (1+4i) gives 4+6i and the magnitude is the distance from the origin, √(4² + 6²) = 7.21. Multiplication behaves differently: magnitudes multiply and angles add, which is why complex numbers describe rotations so naturally. The distinction matters in engineering, where adding two phasors and multiplying them answer entirely different questions.
Frequently Asked Questions
How do you add complex numbers?
Add the real parts and add the imaginary parts separately: (3 + 2i) + (1 + 4i) = 4 + 6i.
Why does multiplication give a real term?
Because i × i = −1, so the product of the imaginary parts becomes real and combines with the real product.
How do you divide complex numbers?
Multiply numerator and denominator by the conjugate of the denominator, which makes the denominator a real number.
What is the conjugate?
The same number with the sign of the imaginary part flipped: the conjugate of 3 + 2i is 3 − 2i.
What is the modulus?
The distance from the origin, √(a² + b²). For 3 + 2i it is about 3.606.
Where are complex numbers actually used?
In AC circuit analysis, signal processing, control systems and any problem involving rotation or oscillation.
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