Question

Find all \(z\) such that \(z^{4}=16 i\). Write the solutions in rectangular form, \(z=a+i b\), with no decimal approximation or trig functions.

Short answer

The solutions are \(z = 2\cos(\frac{\pi}{8})+2i\sin(\frac{\pi}{8}), 2\cos(\frac{5\pi}{8})+2i\sin(\frac{5\pi}{8}), 2\cos(\frac{9\pi}{8})+2i\sin(\frac{9\pi}{8}), 2\cos(\frac{13\pi}{8})+2i\sin(\frac{13\pi}{8}).\)

Step-by-step solution

Represent the number in polar form

Start by converting the number on the right of the given equation, \(16 i\), to polar form: \[16i = 16(\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})).\] Then express it in the form \(re^{i{\Theta}}\), \[16i = 16e^{i\frac{\pi}{2}}.\]

Apply the De Moivre's Theorem

Next, use De Moivre's theorem, which states for a complex number in polar form we have \((r (\cos({\Theta})+i\sin({\Theta})))^n=r^n (\cos(n{\Theta})+i\sin(n{\Theta}))\). The given equation becomes \[z^4 = 16 e^{i\frac{\pi}{2}}\rightarrow z = \sqrt[4]{16} e^{\frac{i\frac{\pi}{2}}{4}+ \frac{2k\pi i}{4}} = 2 e^{\frac{\pi i}{8}+{\frac{k\pi i}{2}}}\] for \(k = 0, 1, 2, 3\) are the 4 different fourth roots.

Transition to rectangular form

Finally, write down the solutions in rectangular form, \(a+bi\). Use the Euler's formula \(e^{i{\Theta}} = \cos({\Theta})+i\sin({\Theta})\). So the four solutions become: \[z = 2(\cos(\frac{\pi}{8}+k\frac{\pi}{2})+i\sin(\frac{\pi}{8}+k\frac{\pi}{2})),\] where \(k = 0, 1, 2, 3\) are the four different solutions.

Key concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in the field of complex numbers, particularly when you want to raise a complex number to a certain power. It's like a superpower for complex numbers that lets you perform what could be a very tricky calculation with relative ease.

For a complex number in polar form, which looks like this: r(cos(θ) + i*sin(θ)), where r is the magnitude and θ is the angle, De Moivre's Theorem states that raising this number to an integer power n is as simple as raising r to the power of n and multiplying the angle θ by n. Mathematically, it's expressed like this:
\[(r(\cos(θ) + i\sin(θ)))^n = r^n(\cos(nθ) + i\sin(nθ))\].

So if you're grappling with a problem like raising a complex number to the fourth power, this theorem spares you from multiplying it out the long way. As shown in the textbook problem, when we apply De Moivre's Theorem, it breaks down a seemingly daunting task into manageable steps, leading us to find the fourth roots of a complex number effortlessly.

Euler's Formula

Euler's Formula might seem like magic at first. It connects the world of trigonometry with complex exponentials using the following beautiful relationship:
\[e^{iθ} = \cos(θ) + i\sin(θ)\].

This elegant equation allows us to shift between the polar and exponential forms of complex numbers seamlessly. When we're faced with complex powers and roots in polar form, Euler's Formula comes to our aid, making it far simpler to convert back and forth from exponential expressions, like in our textbook problem example.

It’s because of this formula that we can represent complex numbers in a more compact form and handle complex calculations like multiplication, division, and even taking roots quite effectively. In essence, Euler’s Formula is not just a neat mathematical trick—it’s a fundamental bridge between exponential functions and trigonometry that greatly simplifies many operations with complex numbers.

Rectangular Form of Complex Number

The rectangular form of a complex number may be the first way you learn to write complex numbers. It literally plots them on a standard, two-dimensional graph, known as the complex plane, with its form being a + bi where a and b are real numbers, and i is the imaginary unit.

In the world of complex numbers, 'a' represents the horizontal (real) axis and 'b' represents the vertical (imaginary) axis. This makes it straightforward to visualize and understand complex numbers, as each one corresponds to a unique point, or coordinate, on the plane.

Transitioning from polar to rectangular form is a cinch with Euler's Formula, as shown in the textbook solution. This is essential for expressing our solution in a way that aligns with real-world applications where a clear, numerical form without trigonometric functions is often required. Rectangular form is the go-to for adding or subtracting complex numbers and it's utterly indispensable when you need to plug complex numbers into real-world situations.