Solving the Magnitude of a Complex Expression with Cube Roots of Unity

The question asks for the value of the magnitude of the expression \(\left|\frac{1-\omega}{\omega+\omega^2}\right|\), where \(\omega\) is a non-real cube root of unity.

Understanding the properties of cube roots of unity is key to solving this problem. The cube roots of unity are the solutions to the equation \(z^3 = 1\). These solutions are \(1, \omega, \omega^2\), where \(\omega\) and \(\omega^2\) are the non-real roots.

Key properties of the cube roots of unity:

• The sum of the cube roots of unity is zero: \(1 + \omega + \omega^2 = 0\).
• The cube of any non-real cube root of unity is one: \(\omega^3 = 1\).

Let's simplify the expression inside the magnitude, \(\frac{1-\omega}{\omega+\omega^2}\).

Using the property \(1 + \omega + \omega^2 = 0\), we can express the denominator \(\omega+\omega^2\) in terms of \(1\):

\[ \omega + \omega^2 = -1 \]

Now, substitute this into the expression:

\[ \frac{1-\omega}{\omega+\omega^2} = \frac{1-\omega}{-1} \]

Simplifying the fraction:

\[ \frac{1-\omega}{-1} = -(1-\omega) = \omega - 1 \]

So, the problem reduces to finding the magnitude of \(\omega - 1\), i.e., \(|\omega - 1|\).

We can calculate this magnitude using the definition of magnitude of a complex number or using properties of \(\omega\).

Method 1: Using the value of \(\omega\)

The non-real cube root of unity \(\omega\) can be written as \(\omega = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\).

Now, calculate \(\omega - 1\):

\[ \omega - 1 = \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) - 1 \] \[ \omega - 1 = \left(-\frac{1}{2} - 1\right) + i\frac{\sqrt{3}}{2} \] \[ \omega - 1 = -\frac{3}{2} + i\frac{\sqrt{3}}{2} \]

The magnitude of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\).

\[ |\omega - 1| = \left|-\frac{3}{2} + i\frac{\sqrt{3}}{2}\right| = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \] \[ |\omega - 1| = \sqrt{\frac{9}{4} + \frac{3}{4}} \] \[ |\omega - 1| = \sqrt{\frac{9+3}{4}} \] \[ |\omega - 1| = \sqrt{\frac{12}{4}} \] \[ |\omega - 1| = \sqrt{3} \]

Method 2: Using properties of complex numbers and \(\omega\)

We want to find \(|\omega - 1|\). The square of the magnitude of a complex number \(z\) is \(z \bar{z}\).

\[ |\omega - 1|^2 = (\omega - 1)(\overline{\omega - 1}) \]

Since the conjugate of a sum/difference is the sum/difference of conjugates, \(\overline{\omega - 1} = \bar{\omega} - \bar{1}\). The conjugate of 1 (a real number) is 1, so \(\bar{1}=1\). Also, for a non-real cube root of unity \(\omega\), its conjugate \(\bar{\omega}\) is equal to \(\omega^2\).

\[ |\omega - 1|^2 = (\omega - 1)(\omega^2 - 1) \]

Expand the product:

\[ |\omega - 1|^2 = \omega(\omega^2) - \omega(1) - 1(\omega^2) + (-1)(-1) \] \[ |\omega - 1|^2 = \omega^3 - \omega - \omega^2 + 1 \] \[ |\omega - 1|^2 = \omega^3 - (\omega + \omega^2) + 1 \]

Using the properties \(\omega^3 = 1\) and \(\omega + \omega^2 = -1\):

\[ |\omega - 1|^2 = 1 - (-1) + 1 \] \[ |\omega - 1|^2 = 1 + 1 + 1 \] \[ |\omega - 1|^2 = 3 \]

Therefore, taking the square root:

\[ |\omega - 1| = \sqrt{3} \]

Both methods confirm that the value of \(\left|\frac{1-\omega}{\omega+\omega^2}\right|\) is \(\sqrt{3}\).

Comparing with the given options, the value \(\sqrt{3}\) corresponds to Option 1.