Free · Complex numbers
Complex Conjugate & Magnitude from Real Components
Find the conjugate of a complex number by reversing its imaginary component, and calculate its nonnegative magnitude.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
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Formula and method
For z = 3 + 4i, the conjugate is 3 − 4i and the magnitude is 5. For z = −3 − 4i, the conjugate is −3 + 4i and the magnitude is still 5.
Conjugate of a + bi = a − bi; magnitude = √(a² + b²)
Worked example
Enter these known values and leave the other values blank.
- Real component a
- 3
- Imaginary coefficient b
- 4
- Magnitude |z|
- 5
Assumptions and limitations
- Enter the real components a and b separately; the imaginary field takes the coefficient, not an expression containing i.
- Magnitude is nonnegative. Recovering a component from magnitude gives the nonnegative square-root branch; magnitude alone cannot identify its sign.
- Enter both signed components directly to obtain the conjugate. The conjugate coefficient is shown in the result sentence.
Common questions
What is the conjugate of a real number?
It is the same real number because its imaginary coefficient is zero.
Does taking the conjugate change the magnitude?
No. Reversing the imaginary sign leaves a² + b² unchanged.
References
A complex number’s magnitude cannot be smaller than the absolute value of either entered real component.