z̄
Complex Conjugate Calculator
Find the conjugate of complex number a + bi
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📐Complex Conjugate Properties
Definition
z̄ = a - bi for z = a + bi
Negate imaginary part
Product Property
z × z̄ = |z|² = a² + b²
Always real and non-negative
Sum Property
z + z̄ = 2a
Always real
Difference Property
z - z̄ = 2bi
Always purely imaginary
💼Applications
Algebra
• Rationalizing denominators
• Complex division
• Finding real/imaginary parts
Engineering
• Signal processing
• Impedance calculations
• Control systems
Physics
• Quantum mechanics
• Wave functions
• Electromagnetic fields
💡Conjugate Rules
• (z̄)̄ = z (double conjugate)
• (z + w)̄ = z̄ + w̄ (sum rule)
• (z × w)̄ = z̄ × w̄ (product rule)
• (z/w)̄ = z̄/w̄ (quotient rule)