The magnitude of the complex conjugate is the same as the magnitude of the original complex number.
Here's why:
- A complex number is represented as a + bi, where a and b are real numbers and i is the imaginary unit (√-1).
- The complex conjugate of a + bi is a - bi.
- The magnitude of a complex number is calculated using the Pythagorean theorem: √(a² + b²).
Therefore, the magnitude of both a + bi and a - bi is √(a² + b²), confirming that the magnitudes are identical.
For example:
- The complex number 3 + 4i has a magnitude of √(3² + 4²) = 5.
- Its conjugate, 3 - 4i, also has a magnitude of √(3² + 4²) = 5.
In essence, the complex conjugate simply flips the sign of the imaginary part, leaving the real part and the magnitude unchanged.
