Problem 86 - Precalculus with Limits

Question

Finding the \(n\) th Roots of a Complex Number \(\operatorname{In}\) Exercises \(81-96,(\) a) use the formula on page 446 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Cube roots of $$-4 \sqrt{2}(-1+i)$$

Step-by-Step Solution

Answer

The cube roots of \(-4\sqrt{2}(-1+i)\) can be found using De Moivre’s Theorem and represent the roots graphically in the Argand plane. The roots should then be represented in standard form \(a + bi\). The exact values depend on the particular computations.

Step 1: Express the given number in the form \(r(\cos \theta + i \sin \theta)\)

Express \( -4\sqrt{2}(-1+i) \) in polar form as \( r\cos\theta + ir\sin\theta \). Here, \( r \) is the magnitude of the complex number and \( \theta \) is the argument. We can find these using the formulas \( r = \sqrt{Re^2 + Im^2} \), \( \theta = \arctan(\frac{Im}{Re}) \).

Step 2: Apply De Moivre’s Theorem to find the cube roots

According to De Moivre’s Theorem, for any complex number in the form \(r(\cos\theta + i\sin\theta)\), the \(m\) th root can be found by \(r^{1/m}(\cos(\frac{θ + 2πk}{m}) + i\sin(\frac{θ + 2πk}{m}))\), where \(m\) is the root we want to find and \(k=[0,1,…,m−1]\). We are looking for the cube roots, so \(m=3\). Applying the formula gives us the three cube roots of the complex number.

Step 3: Represent the roots graphically

Graphical representation of complex roots can be done using an Argand plane, plotting the points corresponding to the three cube roots from step 2.

Step 4: Write roots in standard form

Finally, the roots found in step 2 are to be expressed in standard form, \(a + bi\). These can be obtained by expanding \(r^{1/m}(\cos(\frac{θ + 2πk}{m}) + i\sin(\frac{θ + 2πk}{m}))\).

Key Concepts

nth Roots

When we talk about finding the nth roots of a complex number, we're essentially seeking to find those special numbers that, when raised to the power of n, give us the original complex number. In our exercise, we're interested in the cube roots, which means n equals 3.

In complex analysis, if a complex number is expressed in polar form as \( r(\cos \theta + i \sin \theta) \), its nth roots are determined using the formula:

• \[ r^{1/n}(\cos \left( \frac{\theta + 2\pi k}{n} \right) + i \sin \left( \frac{\theta + 2\pi k}{n} \right)) \]
• where \( k \) takes integer values from 0 up to \( n-1 \).

This formula gives us up to n distinct roots because as we increment \( k \), the angle changes, giving us different roots which are symmetrically distributed over the complex plane. This symmetrical distribution makes every nth root an important player in forming a closed polygonal chain around the origin!

De Moivre's Theorem

De Moivre's Theorem is a cornerstone in working with complex numbers, especially when it comes to taking powers and roots. The theorem provides a straightforward way to raise complex numbers in polar form to a power or find their roots.

Here's how it works: For a complex number in the form \( r(\cos \theta + i \sin \theta) \), the theorem tells us that the m-th power of this number is:

• \[ r^m (\cos(m\theta) + i \sin(m\theta)) \]

When finding the nth roots, we simply rearrange terms and utilize the angle adjustments to unlock smaller powers. This fantastic theorem simplifies calculations, particularly when working with complex exponentials. For cube roots, we take on the transformation of angles in increments of \( \frac{2\pi}{3} \), aligning with the number of roots we're interested in (which is three in this case).

Polar Form

The polar form of a complex number beautifully captures its magnitude and direction, portraying it as a vector in the complex plane. The representation \( r(\cos \theta + i \sin \theta) \) makes this clear.

Here:

• \( r \) is the magnitude, which is the length of the vector from the origin to the point in the Argand plane. It is calculated as \( r = \sqrt{Re^2 + Im^2} \).
• \( \theta \) is the argument, or angle, formed with the positive real axis, calculated using \( \theta = \arctan\left(\frac{Im}{Re}\right) \).

This form simplifies multiplication, division, and particularly finding powers and roots of complex numbers because operations on the magnitude and angle correspond directly to operations in algebra! It shows the elegance of complex numbers in mathematics and provides insightful geometric interpretations.

Standard Form

Standard form is the conventional way of expressing complex numbers as a sum of a real number and an imaginary number, written as \( a + bi \). This form is particularly useful for addition and subtraction of complex numbers, and it directly maps into real and imaginary components.

After finding roots in polar form, the final step usually involves converting back to standard form to express solutions in a more algebra-friendly way.

• This conversion involves evaluating trigonometric components: \( a = r\cos(\theta) \) and \( b = r\sin(\theta) \).
• The combined result is \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary part.

Understanding both polar and standard forms allows for flexibility in computations and interpretations, highlighting the dual nature of complex numbers as algebraic entities and geometric vectors.