Complex Conjugate Calculator

Find the conjugate of a complex number z = a + bi

Enter Complex Number z = a + bi

Original Number (z)

z = 3 + 4i

Complex Conjugate (z*)

z* = 3 - 4i

Other Titles
Understanding the Complex Conjugate
A guide to the definition, properties, and applications of the complex conjugate in mathematics and beyond.

Understanding the Complex Conjugate: A Comprehensive Guide

  • Definition of a complex conjugate
  • Geometric interpretation in the complex plane
  • The simple process of finding the conjugate
The complex conjugate of a complex number is another complex number that has the same real part and an imaginary part of the same magnitude but opposite sign. If a complex number is written as z = a + bi, where 'a' is the real part and 'b' is the imaginary part, its complex conjugate is denoted as z or z̄ and is given by z = a - bi.
Geometric Meaning
In the complex plane, the complex conjugate is the reflection of the original number across the real (horizontal) axis. For example, the number 3 + 4i is located at the point (3, 4). Its conjugate, 3 - 4i, is located at (3, -4), a direct reflection across the x-axis.

Simple Examples

  • If z = 5 + 2i, its conjugate z* = 5 - 2i.
  • If z = 7 - 3i, its conjugate z* = 7 + 3i.
  • If z = -8i (which is 0 - 8i), its conjugate z* = 8i.
  • If z = 4 (which is 4 + 0i), its conjugate z* = 4. Real numbers are their own conjugates.

Step-by-Step Guide to Using the Complex Conjugate Calculator

  • Inputting the real and imaginary components
  • Reading the instantaneous result
This calculator provides an immediate result without a 'calculate' button, updating as you type.
How to Use It

Using the Calculator

  • Enter a = -2 and b = 5. The calculator shows z = -2 + 5i and z* = -2 - 5i.
  • Enter a = 10 and b = -1. The calculator shows z = 10 - 1i and z* = 10 + 1i.

Real-World Applications of the Complex Conjugate

  • Dividing complex numbers
  • Electrical engineering and AC circuits
  • Quantum mechanics
Rationalizing the Denominator
The most important property of the complex conjugate is that when a complex number is multiplied by its conjugate, the result is a real number. Specifically, (a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² + b². This is used to rationalize the denominator when dividing complex numbers, similar to how we rationalize square roots.
Electrical Engineering
In AC circuit analysis, complex numbers (phasors) are used to represent impedance, voltage, and current. The complex conjugate is crucial for calculating apparent power, power factor, and for optimizing power transfer (maximum power transfer theorem).
Quantum Mechanics
Wave functions (ψ), which describe the state of a quantum system, are often complex-valued. The probability of finding a particle in a certain location is given by the product of the wave function and its complex conjugate, |ψ|² = ψ*ψ, which ensures the probability is a real, non-negative number.

Application Example: Division

  • To compute (2 + 3i) / (4 - 5i), we multiply the numerator and denominator by the conjugate of the denominator, which is (4 + 5i):
  • Numerator: (2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i² = 8 + 22i - 15 = -7 + 22i.
  • Denominator: (4 - 5i)(4 + 5i) = 4² + 5² = 16 + 25 = 41.
  • Result: (-7 + 22i) / 41 = -7/41 + (22/41)i.

Common Misconceptions and Correct Methods

  • Changing the sign of the real part
  • Confusing conjugate with negative
Misconception 1: Changing the Wrong Sign
The conjugate of a + bi is a - bi. Only the sign of the imaginary part changes. A common mistake is to change the sign of the real part as well, or instead of the imaginary part. The conjugate of -3 + 2i is -3 - 2i, not 3 - 2i.
Misconception 2: Conjugate vs. Negative
The conjugate z* is different from the negative of z. The negative of z = a + bi is -z = -a - bi. The conjugate only flips the imaginary part's sign, while the negative flips both.

Correct vs. Incorrect

  • For z = -5 - 6i:
  • Correct Conjugate (z*): -5 + 6i
  • Incorrect Conjugate: 5 - 6i or 5 + 6i
  • Correct Negative (-z): 5 + 6i

Mathematical Derivation and Properties

  • Key properties of conjugation
  • Proof of z * z* = |z|²
Properties of Complex Conjugation
Proof: z z is the Squared Modulus
The modulus (or magnitude) of a complex number z = a + bi is its distance from the origin in the complex plane, given by |z| = √(a² + b²).
As shown earlier, z z = (a + bi)(a - bi) = a² + b².
Notice that this is exactly the square of the modulus: |z|² = (√(a² + b²))² = a² + b². Therefore, z z = |z|².

Property Example

  • Let z₁ = 1+i and z₂ = 2-3i.
  • z₁ + z₂ = 3-2i. The conjugate is (z₁ + z₂)* = 3+2i.
  • z₁* = 1-i and z₂* = 2+3i. The sum is z₁* + z₂* = (1-i) + (2+3i) = 3+2i.
  • This verifies that (z₁ + z₂)* = z₁* + z₂*.