Complex Conjugate Calculator
Solve complex conjugate problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
conj(a + bi) = a - bi
The complex conjugate negates the imaginary part while preserving the real part. Key property: z * conj(z) = a^2 + b^2 = |z|^2 (always a non-negative real number). The conjugate reflects the number across the real axis in the complex plane.
Worked Examples
Example 1: Conjugate of 3 + 4i
Problem: Find the complex conjugate of z = 3 + 4i and compute z * conj(z).
Solution: Conjugate: conj(3 + 4i) = 3 - 4i\nProduct: (3 + 4i)(3 - 4i) = 9 - 12i + 12i - 16i^2\n= 9 + 16 = 25\nModulus: |z| = sqrt(25) = 5\nArgument: arctan(4/3) = 53.13 degrees
Result: conj(z) = 3 - 4i | z*conj(z) = 25 | |z| = 5
Example 2: Dividing (2 + 3i) by (1 - 2i)
Problem: Divide (2 + 3i) by (1 - 2i) using the conjugate method.
Solution: Multiply by conjugate of denominator:\n(2+3i)(1+2i) / ((1-2i)(1+2i))\nNumerator: 2 + 4i + 3i + 6i^2 = 2 + 7i - 6 = -4 + 7i\nDenominator: 1 + 4 = 5\nResult: (-4 + 7i)/5 = -0.8 + 1.4i
Result: (2+3i)/(1-2i) = -0.8 + 1.4i
Frequently Asked Questions
What is the complex conjugate and how is it defined?
The complex conjugate of a complex number z = a + bi is denoted as z-bar (or z*) and equals a - bi. It is formed by negating the imaginary part while keeping the real part unchanged. Geometrically, the conjugate is the reflection of z across the real axis in the complex plane (Argand diagram). Every real number is its own conjugate since the imaginary part is zero. The concept of conjugation is fundamental to complex analysis and appears throughout mathematics, physics, and engineering. It preserves the magnitude of the complex number while reversing the direction of rotation in the complex plane.
Why is the product of a complex number and its conjugate always real?
When you multiply z = a + bi by its conjugate a - bi, you get (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2. Since a^2 and b^2 are both real numbers, their sum is always a non-negative real number. This result equals the square of the modulus |z|^2. This property is not a coincidence but follows from the algebraic structure of complex multiplication and the definition of conjugation. It is the key property that makes conjugates useful for rationalizing complex denominators, since multiplying numerator and denominator by the conjugate of the denominator eliminates the imaginary part from the denominator.
How are complex conjugates used to divide complex numbers?
To divide two complex numbers (a + bi) / (c + di), multiply both numerator and denominator by the conjugate of the denominator (c - di). This transforms the denominator into c^2 + d^2 (a real number), making the division straightforward. The result is [(ac + bd) + (bc - ad)i] / (c^2 + d^2). For example, (3 + 2i)/(1 + i) = (3 + 2i)(1 - i)/((1 + i)(1 - i)) = (3 - 3i + 2i - 2i^2)/(1 + 1) = (5 - i)/2 = 2.5 - 0.5i. This technique, called rationalizing the denominator, is analogous to multiplying by the conjugate surd when simplifying expressions with square roots.
What properties do complex conjugates satisfy?
Complex conjugates satisfy several important algebraic properties. The conjugate of a sum equals the sum of conjugates: conj(z1 + z2) = conj(z1) + conj(z2). The conjugate of a product equals the product of conjugates: conj(z1 * z2) = conj(z1) * conj(z2). The conjugate of a conjugate returns the original: conj(conj(z)) = z (involution property). The conjugate of a quotient equals the quotient of conjugates. The real part of z equals (z + conj(z))/2, and the imaginary part equals (z - conj(z))/(2i). These properties make conjugation an automorphism of the complex number field, preserving the algebraic structure of addition and multiplication.
How do complex conjugates appear in polynomial equations?
The Complex Conjugate Root Theorem states that if a polynomial with real coefficients has a complex root a + bi, then its conjugate a - bi must also be a root. This means complex roots of real polynomials always come in conjugate pairs. For example, if x^2 + 1 = 0 has root i, it must also have root -i (which is the conjugate of i). This theorem explains why odd-degree polynomials with real coefficients always have at least one real root, since complex roots pair up and cannot account for an odd number of total roots. The theorem is a direct consequence of the fact that polynomial evaluation commutes with conjugation when coefficients are real.
What is the geometric meaning of complex conjugation in the complex plane?
In the complex plane (Argand diagram), complex conjugation corresponds to reflection across the real axis (the horizontal axis). The number z = a + bi maps to the point (a, b), and its conjugate a - bi maps to (a, -b), which is the mirror image across the x-axis. This reflection preserves the distance from the origin (the modulus remains the same) but reverses the sign of the argument (angle). If z has argument theta, then conj(z) has argument -theta. This geometric interpretation explains why |z * conj(z)| = |z|^2 and why conjugation preserves modulus: both z and its conjugate are equidistant from the origin.