Question

Express each of the following complex numbers in Cartesian form (x + jy):
(1/2)e^(ʲπ)

Answer 1

The Cartesian form of (1/2)e^{j\pi} is -1/2, using Euler's formula to convert the exponential expression to its real and imaginary parts.

Step-by-step explanation:

The expression (1/2)e^{j\pi} can be converted to Cartesian form by recognizing that the exponential form of a complex number includes a magnitude (1/2 in this case) and a phase angle (\pi radians, which corresponds to 180 degrees). Using Euler's formula, e^{j\theta} = cos(\theta) + jsin(\theta), where j is the imaginary unit, you can express the complex exponential as a sum of its real and imaginary parts.

Calculation step: Let's apply Euler's formula to the given complex number:

(1/2)e^{j\pi} = (1/2)(cos(\pi) + jsin(\pi))

Since cos(\pi) = -1 and sin(\pi) = 0, the Cartesian form becomes:

(1/2)(-1 + j(0)) = -1/2