How Many Real Sixth Roots Does 1 Have

How Many Real Sixth Roots Does 1 Have? A Deep Dive into Complex Numbers

The question of how many real sixth roots the number 1 possesses might seem deceptively simple at first glance. A quick mental calculation might lead some to answer "one," pointing to the obvious solution of 1 itself. However, delving deeper into the realm of complex numbers reveals a richer, more nuanced answer. This exploration will unravel the mystery, explaining not only the real roots but also the complex roots, and the underlying mathematical principles that govern their existence.

Understanding Roots and the Fundamental Theorem of Algebra

Before we tackle the specific problem of the sixth roots of 1, let's lay a solid foundation. The nth root of a number x is a number r such that rⁿ = x. For example, the square roots of 9 are 3 and -3 because 3² = 9 and (-3)² = 9.

The Fundamental Theorem of Algebra is crucial to understanding the number of roots a polynomial equation possesses. It states that a polynomial of degree n has exactly n roots (including multiplicities) in the complex number system. This theorem is profound because it encompasses both real and complex numbers. Since finding the nth roots of a number is equivalent to solving the polynomial equation xⁿ - a = 0, where a is the number in question, the Fundamental Theorem of Algebra directly informs us about the number of roots we should expect.

In our case, we are looking for the sixth roots of 1, meaning we need to solve the equation x⁶ - 1 = 0. According to the Fundamental Theorem of Algebra, this equation has exactly six roots.

De Moivre's Theorem: Unlocking the Complex Roots

While simple intuition might suggest only one real root (1), the Fundamental Theorem of Algebra assures us there are six roots in total. To find these roots, we leverage a powerful tool from complex analysis: De Moivre's Theorem. This theorem provides a method for computing powers and roots of complex numbers expressed in polar form.

A complex number z can be represented in polar form as z = r(cos θ + i sin θ), where r is the magnitude (or modulus) and θ is the argument (or angle). De Moivre's Theorem states that for any integer n:

zⁿ = rⁿ(cos(nθ) + i sin(nθ))

To find the sixth roots of 1, we first express 1 in polar form:

1 = 1(cos 0 + i sin 0)

Now, let's apply De Moivre's Theorem to find the roots. We are looking for values of x such that x⁶ = 1. Let x = r(cos θ + i sin θ). Then:

x⁶ = r⁶(cos(6θ) + i sin(6θ)) = 1(cos 0 + i sin 0)

This equation implies:

r⁶ = 1 and 6θ = 0 + 2kπ, where k is an integer.

From r⁶ = 1, we get r = 1 (since we are considering only the positive real root for the magnitude).

From 6θ = 2kπ, we have θ = kπ/3, where k is an integer. This gives us six distinct values of θ for k = 0, 1, 2, 3, 4, 5:

• k = 0: θ = 0 => x = cos 0 + i sin 0 = 1
• k = 1: θ = π/3 => x = cos(π/3) + i sin(π/3) = 1/2 + i√3/2
• k = 2: θ = 2π/3 => x = cos(2π/3) + i sin(2π/3) = -1/2 + i√3/2
• k = 3: θ = π => x = cos π + i sin π = -1
• k = 4: θ = 4π/3 => x = cos(4π/3) + i sin(4π/3) = -1/2 - i√3/2
• k = 5: θ = 5π/3 => x = cos(5π/3) + i sin(5π/3) = 1/2 - i√3/2

For k=6, we obtain the same root as k=0, and so on for any other integer multiple of 6. Therefore, we have found all six roots.

The Real Sixth Roots of 1

Now, let's address the original question: how many real sixth roots does 1 have? Looking at the six roots we found using De Moivre's Theorem, only two of them are purely real:

• 1: The obvious and straightforward solution.
• -1: This is less intuitive but equally valid, as (-1)⁶ = 1.

Therefore, the answer to the question is two. The other four roots are complex numbers, each having both a real and an imaginary component.

Visualizing the Roots in the Complex Plane

The six sixth roots of unity are evenly spaced around the unit circle in the complex plane. This beautiful geometric interpretation provides a powerful visualization of the solution. Each root is separated by an angle of π/3 radians (60 degrees). This symmetry is a direct consequence of De Moivre's Theorem and the nature of complex exponentiation.

Conclusion: Beyond the Obvious

The seemingly simple question of the sixth roots of 1 unveils a fascinating exploration into the world of complex numbers. While the immediate answer might appear to be one, a deeper mathematical analysis, employing the Fundamental Theorem of Algebra and De Moivre's Theorem, reveals the existence of six roots, two of which are real and four are complex. This exercise demonstrates the importance of understanding the intricacies of complex numbers and the elegant interplay between algebra and geometry in solving such problems. The distribution of these roots on the unit circle in the complex plane provides a visually compelling demonstration of the underlying mathematical principles. This exploration underscores the richness and complexity that lies within even seemingly simple mathematical concepts. The journey beyond the obvious answer provides a valuable learning experience in mathematical reasoning and problem-solving.