Question

Calculate (1) \((1+i \sqrt{3})^{10}\) (2) \((1-i)^{1 / 9}\)

Short answer

The short answer is: (1) \((1+i\sqrt{3})^{10}=1024(\cos(\frac{10\pi}{3}) + i\sin(\frac{10\pi}{3}))\) (2) \((1-i)^{1/9}= (\sqrt[\leftroot{-1}\uproot{3}\scriptstyle 9]{2})(\cos(\frac{-\pi}{36}) + i\sin(\frac{-\pi}{36}))\)

Step-by-step solution

1. Convert the complex number into polar form

We convert the complex number \(1+i\sqrt{3}\) into polar form by finding its magnitude (or modulus) and argument (or angle). The magnitude of the complex number \(a+bi\) is \(r = \sqrt{a^2+b^2}\). So for the complex number \(1+i\sqrt{3}\), we have \(r=\sqrt{1^2+(\sqrt{3})^2} = \sqrt{4} = 2\). The argument of the complex number is the angle θ in radians between the complex number and the real axis. θ can be found as follows: \[ \theta = \arctan(\frac{b}{a}) = \arctan(\frac{\sqrt{3}}{1}) = \frac{\pi}{3} \] So, the polar form of \(1+i\sqrt{3}\) is \(2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))\).

2. Apply De Moivre's theorem for \((1+i \sqrt{3})^{10}\)

Using De Moivre's theorem, we can raise a complex number in polar form to any power, such as \((1+i \sqrt{3})^{10}\). De Moivre's theorem states that for any complex number in polar form \(r(\cos(\theta)+i\sin(\theta))\), and a positive integer \(n\), we have: \[ (r(\cos(\theta)+i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta)) \] Applying this theorem to our complex number, we get: \[ (2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})))^{10} = 2^{10}(\cos(\frac{10\pi}{3}) + i\sin(\frac{10\pi}{3})) \] This is equal to \(1024(\cos(\frac{10\pi}{3}) + i\sin(\frac{10\pi}{3}))\).

3. Convert the complex number into polar form for \((1-i)^{1/9}\)

We convert the complex number \(1-i\) into polar form by finding its magnitude (or modulus) and argument (or angle). The magnitude of the complex number \(1-i\) is \(r = \sqrt{1^2+(-1)^2} = \sqrt{2}\). The argument of the complex number is: \[ \theta = \arctan(\frac{b}{a}) = \arctan(\frac{-1}{1}) = -\frac{\pi}{4} \] So, the polar form of \(1-i\) is \(\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))\).

4. Apply De Moivre's theorem for \((1-i)^{1/9}\)

Using De Moivre's theorem again, we can raise a complex number in polar form to any power, such as \((1-i)^{1/9}\). Applying this theorem to our complex number, we get: \[ (\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})))^{\frac{1}{9}} = (\sqrt{2})^{\frac{1}{9}}(\cos(\frac{-\pi}{36}) + i\sin(\frac{-\pi}{36})) \] This is equal to \((\sqrt[9]{2})(\cos(\frac{-\pi}{36}) + i\sin(\frac{-\pi}{36}))\). So, \((1+i\sqrt{3})^{10}=1024(\cos(\frac{10\pi}{3}) + i\sin(\frac{10\pi}{3}))\) and \((1-i)^{1/9}=(\sqrt[\leftroot{-1}\uproot{3}\scriptstyle 9]{2})(\cos(\frac{-\pi}{36}) + i\sin(\frac{-\pi}{36}))\).

Key concepts

Complex Numbers

Complex numbers are an extension of the real numbers, where each number can be expressed in the form:

• \( a + bi \)

where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). In this expression, \( a \) is called the real part, and \( b \) is called the imaginary part. Complex numbers play a crucial role in various fields, such as engineering, physics, and mathematics.
They make it possible to solve equations that have no real solutions, like \( x^2 + 1 = 0 \). Understanding complex numbers is essential for exploring more sophisticated topics like polar form and trigonometric representation.

Polar Form

Polar form represents a complex number based on its magnitude and argument. This format is particularly useful in simplifying multiplication and powers of complex numbers using De Moivre's Theorem. A complex number \( a + bi \) can be transformed into its polar form, \( r (\cos \theta + i \sin \theta) \), where:

• \( r \) is the magnitude, found using \( \sqrt{a^2 + b^2} \).
• \( \theta \) is the argument, the angle made with the positive x-axis, calculated with \( \tan^{-1}(\frac{b}{a}) \).

This transformation helps break down complex problems, making calculations more manageable by turning them into trigonometric problems.

Magnitude and Argument

The magnitude and argument are critical components in understanding the polar form of a complex number.

• **Magnitude (or modulus) ** \( r \) is the distance from the origin to the point \( (a, b) \) in the complex plane, calculated as \( \sqrt{a^2 + b^2} \). It gives a measure of the size of the complex number.
• **Argument** \( \theta \) is the angle formed with the positive x-axis, calculated using \( \arctan(\frac{b}{a}) \). \( \theta \) describes the direction of the complex number in the complex plane.

While the magnitude provides information about the scale of the complex number, the argument defines its orientation, allowing for a full geometric interpretation of complex numbers on the plane.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number, also known as the polar form, uses trigonometric functions to express complex numbers. This form is particularly useful for multiplication, division, and powers of complex numbers.

• It represents the complex number \( a + bi \) as \( r(\cos \theta + i \sin \theta) \).
• This represents the complex number’s position in the complex plane as a point \( (r, \theta) \) in polar coordinates.

The trigonometric form comes into play heavily when using De Moivre's Theorem. This theorem allows you to raise complex numbers to powers succinctly and find roots, making complex arithmetic simpler and more intuitive. The trigonometric form leverages the periodic nature of sine and cosine to solve complex problems with ease.