1. What is a Complex Conjugate?

The complex conjugate of a complex number \( z = a + bi \) is \( \overline{z} = a - bi \). It reflects the complex number across the real axis in the complex plane.

2. How Does the Calculator Work?

The calculator uses the complex conjugate formula:

Where:

• \( a \) — Real part of the complex number
• \( b \) — Imaginary part of the complex number
• \( i \) — Imaginary unit (\( i^2 = -1 \))

Explanation: The calculator simply changes the sign of the imaginary component while keeping the real part unchanged.

3. Importance of Complex Conjugates

Details: Complex conjugates are essential in many areas of mathematics and engineering, including simplifying complex fractions, finding polynomial roots, and in quantum mechanics.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will display both the original number and its conjugate in proper mathematical notation.

5. Frequently Asked Questions (FAQ)

Q1: What happens when you multiply a complex number by its conjugate?
A: The product is always a real number equal to \( a^2 + b^2 \), which is the square of the magnitude of the complex number.

Q2: What's the conjugate of a real number?
A: The conjugate of a real number (where b=0) is the number itself, since there's no imaginary part to change.

Q3: How are conjugates used in division of complex numbers?
A: Multiplying numerator and denominator by the conjugate of the denominator eliminates the imaginary part in the denominator.

Q4: What's the geometric interpretation of conjugation?
A: In the complex plane, conjugation represents reflection across the real (x) axis.

Q5: Are there any special properties of conjugates?
A: Yes, conjugates preserve addition and multiplication: \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \) and \( \overline{z_1 \times z_2} = \overline{z_1} \times \overline{z_2} \).