Complex Conjugate Calculator

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Complex Conjugate

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Absolute value |a+bi|

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Table of Contents

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What is Complex Conjugate?

A complex conjugate keeps the real part of a number and reverses the sign of its imaginary part. If your number is 3 − 4i, its conjugate is 3 + 4i. This calculator performs that sign change and also reports the absolute value, or modulus, of the original number.

Those two answers do different jobs. The conjugate is another complex number; the absolute value is a nonnegative real number measuring distance from zero. Keeping that distinction clear helps when checking algebra, simplifying a complex fraction, or working with coordinates on the complex plane.

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How to Use This Calculator

Write the number in the form a + bi before entering it. Put the real coefficient in Real part (a), then put the signed imaginary coefficient in Imaginary part (b). For 3 − 4i, enter 3 and −4. Enter only the coefficient in the second field, without the letter i.

Choose Calculate and read the two result boxes separately. Complex Conjugate shows the reflected number. Absolute value shows its distance from the origin, rounded to four decimal places when necessary. For a purely imaginary number, enter zero as the real part. For an ordinary real number, enter zero as the imaginary part; leaving a required field blank is different from entering zero.

You can check the operation by taking the conjugate of the answer again. It should return your starting number. Reset clears the input values and result display when you want to begin a separate example.

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Formula Used

For z = a + bi, the conjugate is z̄ = a − bi. The imaginary unit satisfies i² = −1. There is no multiplication or division involved in taking the conjugate itself: only the sign of b changes.

The modulus is |z| = √(a² + b²). The same formula applies to the conjugate because squaring b and −b gives the same value. Two useful checks follow: z + z̄ = 2a and z × z̄ = a² + b². The first removes the imaginary terms by addition; the second removes them through multiplication.

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Variables Explained

Symbol or inputMeaning in this calculator
aReal coefficient. It may be positive, negative, or zero, and it stays unchanged in the conjugate.
bSigned coefficient of i. Enter −4 for an imaginary term of −4i.
iImaginary unit with square −1. It belongs to the displayed expression, not the numeric input.
z̄Conjugate of z, obtained by reversing only the imaginary coefficient.
|z|Modulus of the original number and its conjugate; a real distance, not an imaginary coefficient.
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Worked Example

Suppose z = 3 − 4i. Enter a = 3 and b = −4. Reversing the imaginary sign gives z̄ = 3 + 4i. The modulus calculation is √(3² + (−4)²) = √(9 + 16) = √25 = 5. The two output boxes should therefore show 3 + 4i and 5.

For an independent check, multiply the pair: (3 − 4i)(3 + 4i) = 9 − 16i² = 25. That agrees with |z|². Notice that the middle terms cancel; replacing the conjugate with −3 + 4i would not produce this check.

A second case shows what happens with a negative real coefficient. For −2 + 5i, the conjugate is −2 − 5i, and the modulus is √29, displayed as approximately 5.3852. The negative real part stays negative.

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Why Use This Calculator?

On the complex plane, conjugation reflects a point across the real axis. The point (3, −4) becomes (3, 4), with the same horizontal coordinate and the same distance from the origin. This picture explains both the sign change and the unchanged modulus.

Conjugates are especially useful in division. Multiplying the numerator and denominator of 1/(3 − 4i) by 3 + 4i produces (3 + 4i)/25. The calculator supplies the conjugate, but it does not perform that full fraction simplification for you. Use the complex number calculator when you need a complete arithmetic operation.

A modulus rounded for display is suitable for reading the result. If it will be squared or used in another exact algebraic step, retain the original expression such as √29 instead of treating 5.3852 as exact.

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Common Mistakes

    Do not change both signs: the negative of 3 − 4i is −3 + 4i, while its conjugate is 3 + 4i. Also, −4 is the imaginary coefficient in the first example; entering positive 4 asks about a different number.

    Do not confuse the modulus with |a| + |b|. For 3 − 4i, adding 3 and 4 gives 7, whereas the distance is 5. Finally, a displayed term such as −0i contributes nothing and can be omitted when writing your answer by hand.

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Frequently Asked Questions

Can a real number have a conjugate?

Yes. Write 7 as 7 + 0i. Reversing the zero imaginary coefficient leaves 7, so every real number equals its own conjugate.

What is the conjugate of a purely imaginary number?

For 6i, enter a = 0 and b = 6. Its conjugate is −6i. The two points lie on opposite sides of zero on the imaginary axis, each with modulus 6.

Does taking the conjugate change the size of a number?

It preserves the modulus. The imaginary coordinate changes sign, but its square in a² + b² stays the same. The number itself generally changes unless b is zero.

Is conjugation the same as finding a reciprocal?

No. A reciprocal requires division: for nonzero z, 1/z = z̄/|z|². Conjugation supplies the numerator of that relationship, but the division by the squared modulus is another step.

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Sources & References

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