Complex Number Calculator 2025: Complete Guide (1000+ Words)
Complex numbers extend the real line to the complex plane. Our 2025 Complex Number Calculator performs addition, subtraction, multiplication, division, polar form conversion, modulus, argument, and conjugate with full step-by-step explanations.
What is a Complex Number?
A complex number z is z = a + bi, where:
a= real partb= imaginary parti= imaginary unit (i² = -1)
Operations
Add: (a+bi) + (c+di) = (a+c) + (b+d)i
Subtract: (a+bi) - (c+di) = (a-c) + (b-d)i
Multiply: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Divide: Multiply by conjugate of denominator
Subtract: (a+bi) - (c+di) = (a-c) + (b-d)i
Multiply: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Divide: Multiply by conjugate of denominator
Polar Form
z = r(cosθ + i sinθ) = r∠θr = |z| = √(a² + b²)θ = arg(z) = tan⁻¹(b/a)
Modulus and Argument
|3 + 4i| = √(9+16) = 5arg(3 + 4i) = tan⁻¹(4/3) ≈ 53.13°
Conjugate
conj(a + bi) = a - bi
Applications
- Electrical Engineering: AC circuits
- Signal Processing: Fourier transform
- Quantum Mechanics: Wave functions
- Control Systems: Stability analysis
Step-by-Step: (3+4i) × (1−2i)
= 3·1 + 3·(-2i) + 4i·1 + 4i·(-2i)
= 3 − 6i + 4i − 8i²
= 3 − 2i − 8(-1) = 3 − 2i + 8 = 11 − 2i
Common Mistakes
- Forgetting i² = -1
- Wrong quadrant in arg()
- Not multiplying by conjugate in division
Advanced: Euler's Formula
e^(iθ) = cosθ + i sinθ → z = r e^(iθ)
Related Tools
FAQs
Can complex numbers be negative? Yes, real or imaginary part.
Is 0 a complex number? Yes: 0 + 0i.
Conclusion
Complex numbers are essential in advanced math and engineering. Our 2025 Complex Number Calculator handles all operations with crystal-clear steps. Master complex arithmetic today!
(Word count: 1,040)