Problem 25 Prove that the conjugate of the ... [FREE SOLUTION] (2024)

Short Answer

complex conjugate

Complex conjugates are essential in working with complex numbers. For a complex number in rectangular form like \( z = x + iy \), the complex conjugate is written as \( \bar{z} = x - iy \). The conjugate essentially flips the sign of the imaginary part. This property helps simplify many calculations involving complex numbers.

polar coordinates

Complex numbers can also be represented using polar coordinates. In polar form, a complex number \( z \) is written as \( z = re^{i\theta} \), where \( r \) is the magnitude (distance from the origin) and \( \theta \) is the angle with respect to the positive real axis. This form makes multiplication and division straightforward, as the magnitudes multiply/divide and the angles add/subtract.

rectangular coordinates

The rectangular (or Cartesian) form of a complex number expresses it as \( z = x + iy \), where \( x \) and \( y \) are real numbers. \( x \) is the real part, and \( y \) is the imaginary part. This form is useful for addition and subtraction and corresponds to points on the complex plane as \( (x, y) \).

complex number quotient

To find the quotient of two complex numbers in polar form: \( z_1 / z_2 = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \). The conjugate of the quotient is \( \bar{z_1 / z_2} = \frac{r_1}{r_2} e^{-i(\theta_1 - \theta_2)} \). This is equivalent to dividing the conjugates individually: \( \bar{z_1} / \bar{z_2} = \frac{r_1 e^{-i\theta_1}}{r_2 e^{-i\theta_2}} = \frac{r_1}{r_2} e^{-i(\theta_1 - \theta_2)} \). Hence, \( \bar{z_1 / z_2} = \bar{z_1} / \bar{z_2} \).

complex number product

To multiply two complex numbers in polar form, we use: \( z_1 z_2 = r_1 e^{i\theta_1} r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)} \). The conjugate of the product \( \bar{z_1 z_2} = r_1 r_2 e^{-i(\theta_1 + \theta_2)} \). Separately, \( \bar{z_1} = r_1 e^{-i\theta_1} \) and \( \bar{z_2} = r_2 e^{-i\theta_2} \). Therefore, \( \bar{z_1} \bar{z_2} = r_1 r_2 e^{-i(\theta_1 + \theta_2)} \). This proves that \( \bar{z_1 z_2} = \bar{z_1} \bar{z_2} \).

complex number difference

Using rectangular coordinates, the difference between two complex numbers \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \) is \( z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2) \). The conjugate of the difference is \( \overline{z_1 - z_2} = (x_1 - x_2) - i(y_1 - y_2) \). This matches the difference of their conjugates individually: \( \bar{z_1} - \bar{z_2} = (x_1 - iy_1) - (x_2 - iy_2) = (x_1 - x_2) - i(y_1 - y_2) \). Hence, it proves \( \overline{z_1 - z_2} = \bar{z_1} - \bar{z_2} \).