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Complex Conjugate Calculator

Calculate the complex conjugate z̄ = a − bi, modulus |z| = √(a²+b²), and product z·z̄ for any complex number a + bi.

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Real Part (a)

Imaginary Part (b)

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Conjugate Result

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Conjugate Result—

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What Is a Complex Conjugate?

A complex conjugate is one of the most fundamental operations in complex analysis. For any complex number z = a + bi, where a is the real part and b is the imaginary coefficient, the conjugate is defined as z̄ = a − bi. The operation preserves the real component while reversing the sign of the imaginary component, geometrically reflecting the number across the real axis of the complex plane.

The Three Core Formulas

The conjugate calculator evaluates three interrelated expressions derived from the definition of complex numbers:

Complex Conjugate: z̄ = a − bi
Modulus (Absolute Value): |z| = √(a² + b²)
Product with Conjugate: z · z̄ = a² + b²

Variable Definitions

a (Real Part): The horizontal component of z plotted along the x-axis of the complex plane. It is any real number.
b (Imaginary Part): The coefficient of the imaginary unit i, where i² = −1. It is plotted along the y-axis of the complex plane.
z̄ (Conjugate): The mirror image of z reflected over the real axis, equal to a − bi.
|z| (Modulus): The Euclidean distance from the origin (0, 0) to the point (a, b) in the complex plane.

Mathematical Derivation

The modulus formula follows directly from the Pythagorean theorem. Representing z = a + bi as the point (a, b) in two-dimensional space, the straight-line distance from the origin is √(a² + b²). This geometric interpretation gives the modulus its physical meaning as the magnitude, or absolute value, of the complex number.

The product identity z · z̄ = a² + b² is established by direct multiplication using the difference-of-squares pattern: (a + bi)(a − bi) = a² − abi + abi − b²i². The imaginary cross-terms cancel, and since i² = −1, the term −b²i² becomes +b², yielding the purely real result a² + b². This identity confirms that z · z̄ = |z|² for all complex numbers.

Practical Applications

Dividing Complex Numbers

The most immediate classroom application is rationalizing a complex denominator. To simplify (3 + 4i) ÷ (1 + 2i), multiply numerator and denominator by the conjugate of the denominator (1 − 2i). The denominator becomes 1² + 2² = 5, a real divisor, enabling straightforward computation of the result.

Electrical Engineering and Signal Processing

In AC circuit analysis, electrical impedance is a complex quantity Z = R + jX. Maximum power transfer from source to load occurs when the load impedance equals the complex conjugate of the source impedance. Fourier transforms of real-valued signals obey conjugate symmetry X(−f) = X*(f), a property that halves the computational cost of processing real signals.

Quantum Mechanics

Quantum wave functions ψ(x) are complex-valued. Observable probability densities must be real and non-negative, computed as ψ*(x) · ψ(x) = |ψ(x)|², where ψ* denotes the complex conjugate. The conjugate operation guarantees a physically interpretable result at every step of the calculation.

Complex Conjugate Root Theorem

For any polynomial with real coefficients, non-real roots always appear in conjugate pairs. If (2 + 5i) is a root, then (2 − 5i) must also be a root. This theorem is foundational in algebra, control theory, and digital filter design.

Worked Example

Let z = 5 + 12i. Applying each formula:

Conjugate: z̄ = 5 − 12i
Modulus: |z| = √(25 + 144) = √169 = 13
Product: z · z̄ = 25 + 144 = 169 = 13²

This confirms the universal identity z · z̄ = |z|² for z = 5 + 12i.

Sources

The definitions and properties above follow standard treatments in complex analysis. Authoritative references include the Complex Number Primer: Conjugate and Modulus (Lamar University) and the graduate-level exposition at Complex Conjugate and Norm (Oregon State University). Worked video examples are available at Khan Academy's complex conjugates lesson, and additional derivations appear in the lecture notes Basics of Series and Complex Numbers (University of Wisconsin–Madison).

Reference

Frequently asked questions

What is the complex conjugate of a complex number?

The complex conjugate of z = a + bi is z̄ = a − bi, formed by negating the imaginary part while keeping the real part unchanged. Geometrically, it reflects the number across the real axis in the complex plane. For example, the conjugate of 3 + 4i is 3 − 4i, and the conjugate of −2 − 7i is −2 + 7i. For purely real numbers where b = 0, the conjugate equals the original number itself.

How do you find the complex conjugate step by step?

To find the complex conjugate, identify the real part a and imaginary part b from the form z = a + bi, then write z̄ = a − bi by changing only the sign of b. For z = 6 + 9i, the conjugate is 6 − 9i. For z = −3 − 5i, the conjugate is −3 + 5i. No arithmetic beyond a sign reversal is required. The conjugate calculator automates this by accepting a and b as separate numeric inputs.

What does the modulus of a complex number represent?

The modulus |z| of a complex number z = a + bi equals √(a² + b²) and represents the straight-line distance from the origin to the point (a, b) in the complex plane, exactly as the Pythagorean theorem computes the hypotenuse. It is always a non-negative real number. For z = 3 + 4i, |z| = √(9 + 16) = 5. The modulus satisfies the multiplicative property |z · w| = |z| · |w| for any two complex numbers z and w.

Why is the product of a complex number and its conjugate always a real number?

Multiplying z = a + bi by z̄ = a − bi applies the difference-of-squares identity: (a + bi)(a − bi) = a² − b²i². Since i² = −1, the term −b²i² becomes +b², and the imaginary cross-terms +abi and −abi cancel exactly. The result a² + b² is always a non-negative real number regardless of the values of a and b. This equals |z|², confirming the identity z · z̄ = |z|² universally.

How are complex conjugates used to divide complex numbers?

To divide (p + qi) by (a + bi), multiply both numerator and denominator by the conjugate of the denominator, (a − bi). The denominator becomes the real number a² + b², eliminating all imaginary terms from the divisor. For example, (1 + i) ÷ (2 + 3i), multiplied by (2 − 3i)/(2 − 3i), gives numerator (2 − 3i + 2i − 3i²) = 5 − i over denominator 4 + 9 = 13, yielding the simplified result (5 − i)/13.

What are real-world applications of complex conjugates?

Complex conjugates appear across engineering and physics. In electrical engineering, maximum power transfer occurs when the load impedance equals the source impedance's complex conjugate, a principle central to antenna and amplifier design. In signal processing, real-valued signals satisfy conjugate symmetry X[−k] = X*[k] in the Fourier domain, halving computation. In quantum mechanics, probability densities are computed as ψ*ψ = |ψ|², guaranteeing real non-negative values. The Complex Conjugate Root Theorem also governs polynomial factoring in control theory and filter design.