Calculate the absolute value and an argument for each of the...

Calculate the absolute value and an argument for each of the following complex numbers: $$ \begin{gathered} -3+\mathrm{i} ; \quad-13 ; \quad(1+\mathrm{i})^{17}-(1-\mathrm{i})^{17} ; \quad \mathrm{i}^{4711} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} \\ \frac{1+\mathrm{i} a}{1-\mathrm{i} a}, a \in \mathbb{R} ; \quad \frac{1-\mathrm{i} \sqrt{3}}{1+\mathrm{i} \sqrt{3}} ; \quad(1-\mathrm{i})^{n}, n \in \mathbb{Z} \end{gathered} $$

Calculate the absolute value and an argument for each of the following complex numbers:
$$
\begin{gathered}
-3+\mathrm{i} ; \quad-13 ; \quad(1+\mathrm{i})^{17}-(1-\mathrm{i})^{17} ; \quad \mathrm{i}^{4711} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} \\
\frac{1+\mathrm{i} a}{1-\mathrm{i} a}, a \in \mathbb{R} ; \quad \frac{1-\mathrm{i} \sqrt{3}}{1+\mathrm{i} \sqrt{3}} ; \quad(1-\mathrm{i})^{n}, n \in \mathbb{Z}
\end{gathered}
$$

Key Concepts

Absolute Value

The absolute value of a complex number, also known as its modulus, measures the distance of the complex number from the origin in the complex plane. It is computed as the square root of the sum of the squares of the real and imaginary parts. This concept is key when transitioning between rectangular and polar forms and is essential in determining the magnitude of the number irrespective of its direction.

Argument of a Complex Number

The argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane. It captures the direction of the complex number and is usually measured in radians. Determining the correct argument involves careful consideration of the sign of the real and imaginary parts to identify the appropriate quadrant.

Polar Representation

Polar representation expresses a complex number in the form r*(cos? + i sin?) or equivalently as re^(i?), where r is the absolute value and ? is the argument. This form simplifies the multiplication, division, and exponentiation of complex numbers by converting these operations into arithmetic on magnitudes and angles rather than on rectangular coordinates.

De Moivre's Theorem

De Moivre's theorem is a principle that relates complex numbers and trigonometry, stating that for any complex number expressed in polar form and any integer n, the nth power of the complex number can be computed by raising the modulus to the nth power and multiplying the argument by n. This theorem is particularly useful for calculating powers and roots of complex numbers.

Operations on Complex Numbers in Polar Form

Operations such as multiplication and division become much simpler when complex numbers are expressed in polar form. Multiplication involves multiplying the moduli and adding the arguments, while division involves dividing the moduli and subtracting the arguments. This property is especially beneficial when dealing with complex expressions involving exponentiation and rational functions.

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