Problem 35 - Precalculus : Building Concepts and Connections

Question

Find the square roots of each complex number. Round all numbers to three decimal places. $$i$$

Step-by-Step Solution

Answer

The square roots of \(i\) are \(\pm (0.707 + 0.707i)\), rounded to three decimal places.

Step 1: Convert To Polar Form

Express \(i\) in polar form. Recall that \(i\) is the same as \(0 + 1i\), with real part (Re) as 0 and imaginary part (Im) as 1. Thus its magnitude (r) can be found using the formula \[r = \sqrt{(Re)^2 + (Im)^2}\] which gives \[r = 1\]. The angle (\(\Theta\)), can be found using the formula \(\Theta = atan2(Im, Re)\), atan2 is the function for the tangent operation but taking into account the correct quadrant for the provided x and y values. Since Im is 1 and Re is 0, atan2 will yield \(\Theta = \pi/2\). Therefore, in polar form, \(i\) becomes \(1 cis (\pi/2)\) or \(1\angle \pi/2\) where \(cis \) stands for \(cosine + i*sin\).

Step 2: Calculate Roots

For a complex number in polar form as \(r cis (\Theta)\), the nth roots are given by the formula if \(n = 2\), \[ w_k = \sqrt{r} cis \left(\frac{\Theta + 2k\pi}{n}\right) \] for \(k = 0, 1, 2, ..., n-1\). So, the square roots of \(i = 1\angle \pi/2\) are \[ w_k = \sqrt{1} cis \left(\frac{\pi/2 + 2k\pi}{2}\right) = cis \left(\frac{\pi/4 + k\pi}{2}\right) \] Computing the roots for \(k = 0\) and \(k = 1\) gives the two square roots of \(i\).

Step 3: Convert Back to Rectangular Form

Once the roots in polar form are calculated, convert them back to rectangular form, i.e. \(a + bi\). For each root, let \(r = 1\) and \(\Theta = \pi/4 + k\pi/2\) (obtained from the previous step). The real part (Re) is found by \(r cos (\Theta)\), and the imaginary part (Im) is found by \(r sin (\Theta)\). Thus the square roots of \(i\) are found by calculating these parts for \(k = 0\) and \(k = 1\).

Key Concepts

Polar Form

The polar form of a complex number is a way to express the number using a magnitude and an angle. Instead of representing complex numbers such as \(a + bi\), polar form focuses on an angle \(\Theta\) and a radius \(r\). This form is particularly useful when multiplying or finding roots of complex numbers.
To convert a complex number to polar form, calculate the magnitude \(r\) using \(r = \sqrt{(Re)^2 + (Im)^2}\). This tells us how far the point is from the origin in the complex plane. For our example, the complex number \(i\), the magnitude is \(1\) because \(i\) represents the point \(0, 1\). - The angle \(\Theta\) is determined by the formula \(\Theta = \text{atan2}(Im, Re)\). This ensures that the angle is measured correctly in its respective quadrant in the complex plane. For \(i\), \(\Theta = \pi/2\).
Therefore, in polar form, \(i\) is expressed as \(1 \angle \pi/2\) (which can also be written as \(1 \text{cis} \pi/2\), where "cis" stands for cosine + \(i\) times sine). This form provides a clear and concise way to handle complex numbers for operations such as root calculation.

Square Roots

Finding the square roots of a complex number using its polar form involves some interesting steps. Once the complex number is in polar form, the process to find its roots becomes systematic and involves using the formula for nth roots.
For complex numbers written as \(r \text{cis}(\Theta)\), the formula for finding the nth roots of the number is \(w_k = \sqrt[n]{r} \text{cis}\left(\frac{\Theta + 2k\pi}{n}\right)\). Here, \(k\) represents the number of the root you are calculating, ranging from \(0\) to \(n-1\), where \(n\) is the degree of the root you want to find (for square roots, \(n = 2\)).
Using this formula for \(i = 1 \angle \pi/2\), we calculate its square roots: \

• For \(k = 0\): \(w_0 = 1 \text{cis}(\pi/4)\).
• For \(k = 1\): \(w_1 = 1 \text{cis}(5\pi/4)\).

These roots give us the two square roots of \(i\) when expressed in polar format. Each root corresponds to a unique angle, providing distinctive values.

Rectangular Form

After calculating the square roots of a complex number in polar form, you often convert them back into rectangular form (\(a + bi\)) for further use, because this is generally a more familiar representation.
To convert the polar roots back to rectangular form, compute the real and imaginary parts separately. For each root with a magnitude \(r\) and angle \(\Theta\), use the following:

• Real part (Re): \(r \cos(\Theta)\)
• Imaginary part (Im): \(r \sin(\Theta)\)

For the example of \(i\): - The root \(w_0 = \text{cis}(\pi/4)\) converts to \(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\).
- The root \(w_1 = \text{cis}(5\pi/4)\) converts to \(-\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i\).
This method gives a clear mapping from the polar angle back to the original form, making these expressions simple to understand and use in subsequent calculations.