Perform operations with complex numbers in rectangular and polar forms. Add, subtract, multiply, divide, and convert between forms with detailed step-by-step solutions.
In an AC circuit, the impedance Z = 4 + 3j Ω (rectangular form). What is the impedance in polar form?
Magnitude: r = √(4² + 3²) = √25 = 5 Ω
Phase Angle: θ = tan⁻¹(3/4) = 36.87°
Polar Form: Z = 5∠36.87° Ω
Impedance in polar form makes it easier to analyze series and parallel AC circuits.
Add z₁ = 3 + 2i and z₂ = 1 - 4i.
Step 1: Add real parts: 3 + 1 = 4
Step 2: Add imaginary parts: 2 + (-4) = -2
Result: z₁ + z₂ = 4 - 2i
Complex addition works like vector addition — components add independently.
Multiply z₁ = 2 + 3i and z₂ = 1 - i.
Step 1: Use FOIL: (2)(1) + (2)(-i) + (3i)(1) + (3i)(-i)
Step 2: Simplify: 2 - 2i + 3i - 3i²
Step 3: Since i² = -1: 2 + i + 3 = 5 + i
The product of two complex numbers is another complex number.
Two phasors in polar form: V₁ = 10∠30° V and V₂ = 5∠60° V.
Product: V₁ × V₂ = (10 × 5) ∠ (30° + 60°) = 50∠90° V²
Division: V₁ ÷ V₂ = (10 ÷ 5) ∠ (30° - 60°) = 2∠-30°
In polar form, multiply magnitudes and add angles; divide magnitudes and subtract angles.
A complex number is a number of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as i² = -1. Complex numbers are fundamental to electrical engineering, quantum mechanics, signal processing, control theory, and fluid dynamics.
Remember the powers: i¹ = i, i² = -1, i³ = -i, i⁴ = 1. Higher powers cycle through these four values.
When converting to polar form, the angle θ depends on the quadrant of (a,b). Use atan2(b,a) for correct quadrant.
The conjugate of a+bi is a-bi. Multiplying by the conjugate gives a²+b², a real number. Essential for division.
In polar form, multiply magnitudes and add angles. Divide magnitudes and subtract angles. Much easier than rectangular!
A complex number extends the concept of one-dimensional real numbers to a two-dimensional complex plane by adding an imaginary component. Every complex number can be written as z = a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = -1. The real part is Re(z) = a and the imaginary part is Im(z) = b.
Complex numbers can be visualized as points or vectors on the complex plane (Argand diagram), where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric interpretation leads naturally to the polar form representation: z = r(cos θ + i sin θ) = r∠θ, where r is the magnitude (distance from origin) and θ is the argument (angle from the positive real axis).
Complex numbers are indispensable in modern science and engineering. They simplify the analysis of AC circuits, enable quantum mechanics through wave functions, provide the mathematical foundation for digital signal processing via Fourier transforms, and are essential in control theory for analyzing system stability using pole-zero plots.
Addition and Subtraction: Add or subtract the real parts and imaginary parts independently. For example, (3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i. This is analogous to vector addition in 2D space.
Multiplication: Use the FOIL method (First, Outer, Inner, Last) and substitute i² = -1. For example, (2 + 3i)(1 - i) = 2(1) + 2(-i) + 3i(1) + 3i(-i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i.
Division: Multiply both numerator and denominator by the complex conjugate of the denominator to make the denominator real. For example, (a+bi)/(c+di) = (a+bi)(c-di)/(c²+d²).
To convert from rectangular to polar, calculate the magnitude r = √(a² + b²) and the angle θ = tan⁻¹(b/a). The angle must be adjusted based on the quadrant: if a is negative, add π (180°) to the arctan result. The atan2(b, a) function handles this automatically.
To convert from polar to rectangular, use a = r cos θ and b = r sin θ. Make sure the angle is in radians when using trigonometric functions, or convert from degrees by multiplying by π/180.
Impedance (Z = R + jX) in AC circuits is a complex number. Phasor analysis uses complex numbers to simplify AC circuit calculations, making it possible to apply Ohm's Law to reactive components.
The Fourier Transform, the foundation of all signal processing, naturally produces complex numbers. The real part represents amplitude, while the imaginary part represents phase information of frequency components.
System stability is analyzed using pole-zero plots in the complex s-plane. The location of poles determines whether a system is stable, oscillatory, or unstable. Root locus and Nyquist plots are fundamental tools.
The wave function ψ(x,t) is inherently complex-valued. The Schrödinger equation uses complex numbers, and the probability density is given by |ψ|², the magnitude squared of the complex wave function.
⚠️ Important Note: This Complex Number Calculator is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as circuit design, control system analysis, or scientific research. Always consult a qualified professional for complex number calculations in high-stakes contexts.