Find the conjugate and modulus of complex numbers.
Complex Number (a + bi)
Conjugate Result
3 - 4i
Conjugate (a - bi)
The reflection of the complex number across the real axis.
Modulus |z|5.0000
Original z3 + 4i
Real
Imag
Pink point is the conjugate
About this calculator
Understanding Conjugates
The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. Geometrically, it represents a mirror image across the x-axis (the real axis) on the complex plane.
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Pro Tips
The product of a number and its conjugate is always a non-negative real number (a² + b²).
Conjugates are used to rationalize denominators in complex division.
If the imaginary part is zero, the number and its conjugate are identical.
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Fun Facts
"The modulus |z| is the distance from the origin (0,0) to the point (a, b)."
"Complex numbers are essential for describing wave functions in quantum mechanics."
"The sum of a number and its conjugate is twice the real part (2a)."
