How this calculator works
Conjugation reflects a point across the real axis, preserving its magnitude. To find a missing original component from a magnitude, use the Pythagorean relation and explicitly choose the sign. The result always describes the conjugate of the completed original number, including its phase where defined.
Formula
conjugate(a + bi) = a - bi; |z| = sqrt(a^2 + b^2). Missing component = +/- sqrt(|z|^2 - known^2).
Worked example
The conjugate of 3 + 4i is 3 - 4i and both magnitudes are 5. If a = 5 and magnitude is 13, the original imaginary part may be +12 or -12; its conjugate has the opposite sign.
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Frequently asked questions
Does conjugation change a real number?
No. When the imaginary part is zero, the conjugate is the original real number.
Can I recover a unique component from magnitude?
Only after choosing its sign, except when the missing component is zero.
Why is the phase absent at zero?
The zero vector has no direction, so its complex argument is undefined.