Simplify Complex Roots Of Unity Expression

When delving into the fascinating world of complex numbers, you'll inevitably encounter the concept of roots of unity. These are numbers that, when raised to a specific power, equal 1. Among these, the cubic roots of unity hold a special place, representing the solutions to the equation $z^3 = 1$. While 1 is one cubic root of unity, the other two are non-real and have a peculiar, yet incredibly useful, relationship. These are often denoted by $\omega$ and $\omega^2$. Understanding their properties is key to simplifying expressions involving them, like the one we're about to tackle.

Let's dive into the mathematical puzzle presented: If $\omega, \omega^2$ are the non-real cubic roots of unity, what is the value of $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}$? This problem requires us to leverage the fundamental properties of these complex roots. The first and most crucial property is that the sum of all cubic roots of unity is zero: $1 + \omega + \omega^2 = 0$. This implies that $\omega + \omega^2 = -1$. Another vital property is that $\omega^3 = 1$. Furthermore, $\omega^2 = \frac{1}{\omega}$ and $(\omega^2)^2 = \omega^4 = \omega$. These relationships are the building blocks for simplifying any expression involving $\omega$ and $\omega^2$. Without a firm grasp of these properties, attempting to solve such problems can feel like navigating a maze blindfolded. However, with these tools at our disposal, the path to the solution becomes clear and, dare I say, quite elegant.

Unpacking the Properties of Cubic Roots of Unity

Before we fully engage with the given expression, let's reinforce our understanding of the non-real cubic roots of unity. Recall that the cubic roots of unity are the solutions to $z^3 - 1 = 0$. Factoring this equation, we get $(z-1)(z^2+z+1) = 0$. One root is clearly $z=1$. The other two roots come from the quadratic equation $z^2+z+1 = 0$. Using the quadratic formula, $z = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}$. These are our $\omega$ and $\omega^2$. Let's assign $\omega = \frac{-1 + i\sqrt{3}}{2}$ and $\omega^2 = \frac{-1 - i\sqrt{3}}{2}$. If we square $\omega$, we get $(\frac{-1 + i\sqrt{3}}{2})^2 = \frac{1 - 2i\sqrt{3} + (i\sqrt{3})^2}{4} = \frac{1 - 2i\sqrt{3} - 3}{4} = \frac{-2 - 2i\sqrt{3}}{4} = \frac{-1 - i\sqrt{3}}{2}$, which is indeed $\omega^2$. Conversely, squaring $\omega^2$ gives us $\omega^4 = \omega$. This reciprocal relationship is fundamental. More importantly, as we noted earlier, $1 + \omega + \omega^2 = 0$. This identity is incredibly powerful because it allows us to substitute $-1$ for $\omega + \omega^2$, or to express one root in terms of the others, for example, $\omega^2 = -1 - \omega$. These relationships are not just abstract mathematical curiosities; they are the keys that unlock the simplification of complex algebraic expressions involving these roots.

Simplifying the First Term

Let's focus on the first term of the expression: $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}$. Our goal is to simplify this fraction. We can try to manipulate the denominator to resemble the numerator or vice versa, using the properties we've discussed. Let's consider the denominator, $c+a \omega+b \omega^2$. Can we relate this to the numerator $a+b \omega+c \omega^2$? A common strategy when dealing with $\omega$ and $\omega^2$ is to multiply by powers of $\omega$. Let's try multiplying the denominator by $\omega$: $\omega(c+a \omega+b \omega^2) = c\omega + a\omega^2 + b\omega^3$. Since $\omega^3 = 1$, this becomes $c\omega + a\omega^2 + b$. This doesn't immediately look like the numerator. Let's try multiplying by $\omega^2$: $\omega^2(c+a \omega+b \omega^2) = c\omega^2 + a\omega^3 + b\omega^4$. Using $\omega^3 = 1$ and $\omega^4 = \omega$, this simplifies to $c\omega^2 + a + b\omega$. Rearranging the terms, we get $a + b\omega + c\omega^2$. Eureka! The denominator, when multiplied by $\omega^2$, becomes exactly the numerator. This means that $c+a \omega+b \omega^2 = \frac{a+b \omega+c \omega^2}{\omega^2}$. Therefore, the first term simplifies to $\frac{a+b \omega+c \omega^2}{\frac{a+b \omega+c \omega^2}{\omega^2}} = \omega^2$. This is a significant simplification and shows the power of working with these roots. It's crucial to ensure that the denominator is not zero. If $a+b \omega+c \omega^2 = 0$, then $a = b=c=0$ for the equality to hold for all cases. Assuming $a, b, c$ are not all zero, this simplification is valid.

Simplifying the Second Term

Now, let's turn our attention to the second term: $\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}$. We'll employ a similar strategy. Let the numerator be $N = a+b \omega+c \omega^2$. Consider the denominator $D_2 = b+c \omega+a \omega^2$. Let's see what happens when we multiply $D_2$ by $\omega$: $\omega(b+c \omega+a \omega^2) = b\omega + c\omega^2 + a\omega^3$. Using $\omega^3 = 1$, this becomes $b\omega + c\omega^2 + a$. Rearranging, we get $a + b\omega + c\omega^2$, which is our numerator $N$. So, $b+c \omega+a \omega^2 = \frac{a+b \omega+c \omega^2}{\omega}$. This implies that the second term simplifies to $\frac{a+b \omega+c \omega^2}{\frac{a+b \omega+c \omega^2}{\omega}} = \omega$. Again, this simplification is valid as long as the numerator is not zero. The elegance here is striking; we've managed to reduce both complex fractions to simple powers of $\omega$. This highlights the symmetric and cyclic nature of the roots of unity. The structure of the denominators is directly related to cyclic permutations of the coefficients in the numerator, a pattern that $\omega$ and $\omega^2$ are perfectly designed to simplify.

The Grand Finale: Summing the Simplified Terms

We have successfully simplified the two terms of the original expression. The first term, $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}$, reduced to $\omega^2$. The second term, $\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}$, reduced to $\omega$. Now, we just need to add these results together: $\omega^2 + \omega$. Recall the fundamental property of cubic roots of unity: $1 + \omega + \omega^2 = 0$. From this identity, we can directly deduce that $\omega + \omega^2 = -1$. Therefore, the sum of the two simplified terms is -1. This result is independent of the values of $a$, $b$, and $c$, provided that the denominators in the original expression are not zero. The problem elegantly resolves to a fundamental identity of the cubic roots of unity. This demonstrates how seemingly complex algebraic expressions can be unraveled using the inherent properties of mathematical constants and relationships. The beauty lies in the reduction to a simple, universally true statement about these roots.

Let's double-check our steps to ensure accuracy. We used the properties $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$. For the first term, we showed that $\omega^2 (c+a \omega+b \omega^2) = a+b \omega+c \omega^2$. Thus, $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2} = \omega^2$. For the second term, we showed that $\omega (b+c \omega+a \omega^2) = a+b \omega+c \omega^2$. Thus, $\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2} = \omega$. Summing them gives $\omega^2 + \omega = -1$. The solution seems robust. This type of problem is common in algebra and complex numbers, testing a student's grasp of fundamental properties and their application in simplification. The structure of the expression is specifically designed to exploit these properties, leading to a surprisingly simple answer.

Conclusion

In conclusion, by carefully applying the fundamental properties of the non-real cubic roots of unity, namely $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$, we were able to simplify the given expression. The first term $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}$ simplifies to $\omega^2$, and the second term $\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}$ simplifies to $\omega$. Adding these results, we get $\omega^2 + \omega$, which, from the identity $1 + \omega + \omega^2 = 0$, equals -1. Therefore, the value of the entire expression is -1. This problem serves as a beautiful illustration of how abstract mathematical properties can lead to elegant and straightforward solutions for seemingly intricate algebraic challenges. It underscores the importance of mastering the foundational concepts in mathematics.

For further exploration into the fascinating properties of complex numbers and roots of unity, you might find the resources at Brilliant.org to be incredibly helpful and engaging.