5
2
5.385165
—
deg
5.830952
3.605551
0.928477
0.371391
5
2
5.385165
—
deg
5.830952
3.605551
0.928477
0.371391
The Complex Number Addition Calculator computes the sum of two complex numbers, providing the result in both rectangular (a + bi) and polar (magnitude and argument) forms. Addition is the simplest and most intuitive operation on complex numbers, and understanding it thoroughly provides the foundation for all other complex arithmetic operations.
Complex number addition works by adding the real parts together and the imaginary parts together, independently. Given z₁ = a + bi and z₂ = c + di, their sum is z₁ + z₂ = (a + c) + (b + d)i. This rule follows directly from the algebraic structure of complex numbers: they form a field (and specifically a vector space over the reals), where addition is defined component-wise.
Geometrically, complex number addition is identical to vector addition in the complex plane. If z₁ is represented as the point (a, b) and z₂ as the point (c, d), then their sum corresponds to the point (a + c, b + d). This is the familiar parallelogram law: place the two vectors tail-to-tail, complete the parallelogram, and the diagonal is the sum. This visual interpretation makes complex addition intuitive and connects it to physical applications involving superposition of quantities.
The triangle inequality is a fundamental property of complex addition: |z₁ + z₂| ≤ |z₁| + |z₂|. The magnitude of the sum never exceeds the sum of the magnitudes. Equality holds if and only if the two complex numbers have the same argument (point in the same direction). This inequality has deep consequences in analysis, metric space theory, and functional analysis.
In electrical engineering, complex addition is used to combine AC signals represented as phasors. When two alternating currents flow through the same conductor, their combined effect is the sum of their phasor representations. The real part represents the in-phase component, and the imaginary part represents the quadrature (90° out-of-phase) component. Adding phasors is how engineers compute total voltage, current, and impedance in AC circuits.
In quantum mechanics, the superposition principle states that if ψ₁ and ψ₂ are valid quantum states, then any complex linear combination α ψ₁ + β ψ₂ is also a valid state. Complex addition is thus the mathematical expression of one of the most fundamental principles in physics. Interference patterns — both constructive and destructive — arise from the addition of complex amplitudes.
In signal processing, the discrete Fourier transform (DFT) involves summing complex exponentials. Each frequency component of a signal is a complex number, and the total signal is their sum. Complex addition is performed millions of times per second in digital audio, image processing, and communications systems.
Given z₁ = a + bi and z₂ = c + di:
Addition formula:
$$z_1 + z_2 = (a + c) + (b + d)i$$
The real parts add: Re(z₁ + z₂) = a + c
The imaginary parts add: Im(z₁ + z₂) = b + d
Magnitude of the sum:
$$|z_1 + z_2| = \sqrt{(a+c)^2 + (b+d)^2}$$
Argument of the sum:
$$\arg(z_1 + z_2) = \arctan\left(\frac{b+d}{a+c}\right)$$
(using atan2 for correct quadrant determination)
Triangle inequality:
$$|z_1 + z_2| \leq |z_1| + |z_2|$$
The real part of the sum is simply the sum of the two real parts. The imaginary part is the sum of the two imaginary parts. The magnitude of the result represents the distance from the origin to the sum point in the complex plane. Note that this is generally NOT the sum of the individual magnitudes (unless the numbers have the same argument). The argument gives the angle of the sum from the positive real axis. The individual magnitudes of z₁ and z₂ are shown for comparison with the triangle inequality.
Inputs
Results
(3 + 5i) + (2 − 3i) = (3 + 2) + (5 + (−3))i = 5 + 2i. Magnitude = √(25 + 4) = √29 ≈ 5.385. Triangle inequality check: 5.385 ≤ 5.831 + 3.606 = 9.437. ✓
Inputs
Results
(4 + 7i) + (4 − 7i) = 8 + 0i = 8. Adding a complex number and its conjugate always gives a real number: z + z̄ = 2·Re(z). The imaginary parts cancel exactly.
Yes. z₁ + z₂ = z₂ + z₁ for all complex numbers. This follows from the commutativity of real addition applied to both the real and imaginary components independently.
The imaginary parts cancel: (a + bi) + (a − bi) = 2a. The result is always a real number equal to twice the real part. Similarly, subtracting gives z − z̄ = 2bi, a purely imaginary number equal to twice the imaginary part times i.
Because the angle between the two complex numbers matters. By the triangle inequality, |z₁ + z₂| ≤ |z₁| + |z₂|. Equality holds only when both numbers have the same argument (point in the same direction). When they point in opposite directions, the magnitudes partially cancel, and |z₁ + z₂| can be as small as ||z₁| − |z₂||.
Complex addition is mathematically identical to 2D vector addition. The complex number a + bi corresponds to the vector (a, b), and adding two complex numbers gives the same result as adding their corresponding position vectors using the parallelogram rule.
The identity element is 0 + 0i (or simply 0). Adding zero to any complex number leaves it unchanged: z + 0 = z. The additive inverse of z = a + bi is −z = −a − bi, satisfying z + (−z) = 0.
In AC circuit analysis, voltages and currents are represented as phasors (complex numbers). When two AC signals combine, their phasor sum gives the resulting signal. For example, two currents I₁ = 3 + 4i and I₂ = 1 − 2i combine to I_total = 4 + 2i. The magnitude gives the peak current, and the argument gives the phase angle.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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