Complex Conjugate Calculator
Enter the real part and imaginary part of a complex number z = a + bi. The calculator returns the complex conjugate z-bar = a - bi, the modulus |z|, the modulus squared |z|^2 (which equals the product z times z-bar), the argument (angle) in radians or degrees, and the polar form. A step-by-step working panel shows exactly how each result is derived.
What is the complex conjugate?
For any complex number z = a + bi, where a and b are real numbers and i is the imaginary unit (satisfying i^2 = -1), the complex conjugate (written z-bar or z*) is obtained by changing the sign of the imaginary part: z-bar = a - bi. If z = 3 + 4i, then z-bar = 3 - 4i. The real part stays the same; only the imaginary part is negated. Geometrically on the complex plane (Argand diagram), the conjugate is the mirror image of the original point reflected across the horizontal real axis.
How to calculate the complex conjugate
The rule is simple: keep the real part unchanged and flip the sign of the imaginary part. For z = a + bi, write z-bar = a - bi. For z = a - bi (negative imaginary part), write z-bar = a + bi. For a purely real number such as z = 5, the conjugate is 5 itself because b = 0 and negating 0 changes nothing. For a purely imaginary number such as z = 3i, the conjugate is -3i. To use this calculator, type the real part a and the imaginary part b (just the coefficient, not the "i"), and all derived quantities update instantly.
Why the complex conjugate matters - key applications
The conjugate is central to complex number arithmetic and appears throughout mathematics, physics, and engineering. Dividing complex numbers: to compute (p + qi) / (a + bi), multiply numerator and denominator by a - bi (the conjugate of the denominator). The denominator becomes a^2 + b^2, a real number, eliminating the imaginary part from below the fraction line. Finding the modulus: |z|^2 = z times z-bar = a^2 + b^2, so the modulus is the square root of that product. Signal processing: in AC circuit analysis and Fourier transforms, conjugate symmetry means the transform of a real signal has conjugate-symmetric frequency components. Quantum mechanics: the inner product of wavefunctions uses complex conjugation, and observable quantities require the conjugate to produce real expectation values.
Modulus, argument and polar form
Three additional quantities fully characterize a complex number alongside its conjugate. The modulus |z| = sqrt(a^2 + b^2) is the Euclidean distance from the origin to the point (a, b) in the complex plane. Both z and its conjugate z-bar share the same modulus, which explains why they lie at the same distance from the origin (just on opposite sides of the real axis). The argument theta = atan2(b, a) is the angle the line from the origin to z makes with the positive real axis. The argument of z-bar is -theta, the reflection of the original angle. Together, modulus and argument define the polar form: z = r * (cos(theta) + i*sin(theta)), also written r*e^(i*theta) via Euler's formula. The polar form makes multiplication and exponentiation of complex numbers much easier than the rectangular form.
Common complex conjugate identities
| Property | Formula | What it means |
|---|---|---|
| Conjugate definition | conj(a + bi) = a - bi | Negate the imaginary part |
| Double conjugate | conj(conj(z)) = z | Applying twice returns the original |
| Modulus squared | z * conj(z) = |z|^2 = a^2 + b^2 | Product is always real and non-negative |
| Sum | conj(z + w) = conj(z) + conj(w) | Conjugate distributes over addition |
| Product | conj(z * w) = conj(z) * conj(w) | Conjugate distributes over multiplication |
| Quotient | conj(z / w) = conj(z) / conj(w) | Conjugate distributes over division |
| Real numbers | conj(a) = a (when b = 0) | Real numbers are their own conjugate |
| Pure imaginary | conj(bi) = -bi | Purely imaginary number negates |
| Reciprocal | 1/z = conj(z) / |z|^2 | Express inverse using conjugate |
Key algebraic properties that hold for any complex numbers z and w.
Frequently asked questions
What is the complex conjugate of a real number?
A real number has no imaginary part, so b = 0. The conjugate is a - 0i = a, which is exactly the same as the original. Every real number is its own complex conjugate. This is why real numbers appear on the real axis of the complex plane with no mirror image below it.
What is the complex conjugate of a purely imaginary number?
For z = bi (where a = 0), the conjugate is z-bar = -bi. The real part (zero) is unchanged, and the imaginary part is negated. For example, the conjugate of 7i is -7i, and the conjugate of -2i is +2i.
Why do we multiply by the conjugate when dividing complex numbers?
To divide (p + qi) by (a + bi), multiply both numerator and denominator by the conjugate a - bi. The denominator (a + bi)(a - bi) = a^2 + b^2, which is a plain real number. Dividing by a real number is straightforward arithmetic. This technique is called rationalizing the denominator and is the standard method for simplifying complex fractions.
Does the conjugate change the modulus (absolute value)?
No. The modulus |z| = sqrt(a^2 + b^2) and the modulus of the conjugate |z-bar| = sqrt(a^2 + (-b)^2) = sqrt(a^2 + b^2) are identical. Conjugation reflects the point across the real axis, preserving its distance from the origin. Both z and z-bar lie on the same circle of radius |z| centered at the origin.
What does conjugate mean geometrically on the complex plane?
On an Argand diagram (the complex plane), plotting z = a + bi gives a point at coordinates (a, b). Its conjugate z-bar = a - bi sits at (a, -b), the mirror image reflected across the horizontal real axis. The argument (angle from the positive real axis) of z-bar is the negative of the argument of z. This reflection symmetry is why conjugate pairs appear together as roots of polynomials with real coefficients.
If a polynomial has real coefficients, can its roots include complex numbers?
Yes, but only in conjugate pairs. The complex conjugate root theorem states that if z = a + bi (with b not equal to 0) is a root of a polynomial whose coefficients are all real, then z-bar = a - bi must also be a root. This is why complex roots of real polynomials always appear in pairs, and why polynomials of odd degree with real coefficients always have at least one real root.
What is the notation for a complex conjugate?
The most common notations are a bar over the letter (z-bar, written with an overline) and a star superscript (z*). Both mean the same thing: negate the imaginary part. Physics and engineering tend to use z*, while pure mathematics more often uses z-bar with the overline. This calculator uses z-bar in its labels.