Complex Conjugate Calculator
Solve complex conjugate problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy