Question

For any integer \(\mathrm{n}, \arg \left[(\sqrt{3}+\mathrm{i})^{4 n+1} /(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}\right]=\) (a) \((\pi / 3)\) (b) \((\pi / 6)\) (c) \((2 \pi / 3)\) (d) \((5 \pi / 6)\)

Short answer

The principal argument of \(\frac{(\sqrt{3}+\mathrm{i})^{4 n+1}}{(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}}\) is \(\boxed{\pi/3}\). Hence, the correct choice is (a) \((\pi / 3)\).

Step-by-step solution

Simplifying the powers of complex numbers

We first simplify the powers of complex numbers in the numerator and denominator using De Moivre's theorem: \((\cos\varphi + i\sin\varphi)^n = \cos(n\varphi)+ i\sin(n\varphi)\) So, let's first find the polar form of the given complex numbers: \(\sqrt{3} + \mathrm{i}=\) 2\((\frac{\sqrt{3}}{2} + \frac{1}{2} \mathrm{i}) = 2(\cos(\pi/3) + \mathrm{i}\sin(\pi/3))\) \(1-\mathrm{i}\sqrt{3} =\) 2\((\frac{1}{2} - \mathrm{i}\frac{\sqrt{3}}{2}) = 2(\cos(5\pi/3) + \mathrm{i}\sin(5\pi/3))\) Now, we can use De Moivre's theorem to simplify the powers: \((\sqrt{3}+\mathrm{i})^{4 n+1} = 2^{4n+1} (\cos((4n+1)\pi/3) + \mathrm{i}\sin((4n+1)\pi/3))\) \((1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}} = 2^{4n} (\cos(20n\pi/3) + \mathrm{i}\sin(20n\pi/3))\)

Simplifying the fraction

Now we need to simplify the fraction inside the argument function: \(\frac{(\sqrt{3}+\mathrm{i})^{4 n+1}}{(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}}= \frac{2^{4n+1}(\cos((4n+1)\pi/3) + \mathrm{i}\sin((4n+1)\pi/3))}{2^{4n}(\cos(20n\pi/3) + \mathrm{i}\sin(20n\pi/3))}\) The coefficients (powers of 2) will cancel out, and we get: \(\frac{\cos((4n+1)\pi/3) + \mathrm{i}\sin((4n+1)\pi/3)}{\cos(20n\pi/3) + \mathrm{i}\sin(20n\pi/3)}\)

Finding the principal argument

To find the principal argument of the given expression, we can simply subtract the argument of the denominator from that of the numerator: \((4n+1)\pi/3 - 20n\pi/3 = (4n - 20n+1)\pi/3 = -16n\pi/3 + \pi/3\) Since n is an integer, -16nπ/3 represents a multiple of 2π, and thus it won't contribute to the principal argument. So, the principal argument of the expression is: \(\arg\left[\frac{(\sqrt{3}+\mathrm{i})^{4 n+1}}{(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}\right] = \boxed{\pi/3}\) Hence, the correct choice is (a) \((\pi / 3)\).

Key concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool used to work with complex numbers raised to a power. It helps simplify the calculation of powers for complex numbers expressed in polar form.
In mathematical terms, for a complex number expressed as \((\cos \varphi + i \sin \varphi)^n\), De Moivre’s Theorem states:

• \((\cos \varphi + i \sin \varphi)^n = \cos(n\varphi) + i\sin(n\varphi)\)

This means that raising a complex number in polar form to a power \(n\) involves multiplying the argument by \(n\) while keeping the modulus raised to the power \(n\).
This theorem greatly simplifies many problems involving powers of complex numbers, making it easier to manipulate and calculate complex expressions.

Polar Form

Every complex number can be represented in polar form, which highlights its angle and magnitude. This form uses the notation \(r(\cos \theta + i\sin \theta)\), where:

• \(r\) is the modulus (or magnitude) of the complex number, calculated as \(\sqrt{x^2 + y^2}\)
• \(\theta\) is the argument (or angle), which indicates the direction of the vector in the complex plane

By conversion to polar form, calculations like multiplication, division, and finding powers become more straightforward.
For example, the complex number \(\sqrt{3} + i\) can be converted to polar form as \(2(\cos (\pi/3) + i \sin (\pi/3))\). This reflects its position in the complex plane more intuitively, relating to its geometric representation as a vector.

Principal Argument

The principal argument of a complex number is the angle between the positive real axis and the line representing the complex number in the complex plane.
It is typically expressed in radians and falls within the interval \((-\pi, \pi]\). The principal argument helps define the direction of the complex number vector.

• The formula for finding the principal argument \(\arg(z)\) involves trigonometric functions:
• \(\text{If } z = a + bi, \text{ then } \tan \theta = \frac{b}{a}\), where \(\theta\) is the argument.

In problems, such as in the original exercise, the principal argument is calculated by subtracting the argument of the denominator from the argument of the numerator, simplifying complex fractions and ensuring the result fits within \((-\pi, \pi]\).
This calculation is crucial in expressing the "direction" of a result correctly and succinctly.

Complex Fractions

Complex fractions involve the division of two complex numbers. The process is made easier by converting the complex numbers into their polar forms.
The division of two complex numbers, \(\frac{z_1}{z_2}\), is performed by dividing their moduli and subtracting their arguments:

• \(\frac{r_1(\cos \theta_1 + i\sin \theta_1)}{r_2(\cos \theta_2 + i\sin \theta_2)} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))\)

Thus, the modulus is divided, and the angle is adjusted by subtraction.
For instance, when simplifying expressions like \(\frac{(\sqrt{3} + i)^{4n+1}}{(1 - i\sqrt{3})^{4n}}\), the polar form allows us to focus solely on managing the angles and individual magnitudes, simplifying the problem into more manageable terms.