Evaluate the following complex numbers and express your...

Evaluate the following complex numbers and express your results in rectangular form:
(a) $2+\frac{3+j 4}{5-j 8}$
(b) $4 \angle-10^{\circ}+\frac{1-j 2}{3 \angle 6^{\circ}}$
(c) $\frac{8 \angle 10^{\circ}+6 \angle-20^{\circ}}{9 \angle 80^{\circ}-4 \angle 50^{\circ}}$

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are typically written in the form a + jb (or a + bi) where 'a' is the real component and 'jb' (or 'bi') is the imaginary component. Complex numbers allow for the analysis and solution of equations that do not have real solutions and play a crucial role in many areas of engineering and mathematics.

Rectangular Form

The rectangular form of a complex number expresses it in terms of its real and imaginary components, as in a + jb. This form is especially useful for addition and subtraction of complex numbers, as it allows for the direct combination of like terms (i.e., real parts with real parts and imaginary parts with imaginary parts).

Polar Form

The polar form of a complex number expresses it in terms of its magnitude (or modulus) and angle (or argument), commonly written as r ? ?. This representation is particularly useful for multiplication, division, and finding powers and roots of complex numbers because the operations become simpler when applied to the magnitude and angle separately.

Conversion Between Forms

Converting between the rectangular and polar forms of a complex number involves using trigonometric relationships. The conversion from rectangular to polar form uses the relationships r = ?(a² + b²) and ? = arctan(b/a), while conversion from polar to rectangular form uses a = r cos? and b = r sin?. This conversion is essential when different operations (like multiplication or division) are more easily performed in one form than the other.

Complex Arithmetic Operations

Operations with complex numbers—including addition, subtraction, multiplication, and division—require careful consideration of their representation. Addition and subtraction are most straightforward in rectangular form, while multiplication and division can be easier in polar form. For division in rectangular form, one typically multiplies the numerator and denominator by the complex conjugate of the denominator to simplify the expression and eliminate any complex number from the denominator.

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