Problem 1 - Algebra and Geometry

Question

Show that the three cube roots of unity are \(1,(-1+i \sqrt{3}) / 2\) and \((-1-i \sqrt{3}) / 2\)

Step-by-Step Solution

Answer

The three cube roots of unity are 1, \(\frac{-1 + i \sqrt{3}}{2}\), and \(\frac{-1 - i \sqrt{3}}{2}\).

Step 1: Define Cube Roots of Unity

The cube roots of unity are the solutions to the equation \(x^3 = 1\). These roots are also the solutions of the equation converted to a polynomial form: \(x^3 - 1 = 0\).

Step 2: Factor the Polynomial

Rewrite \(x^3 - 1 = 0\) as \((x - 1)(x^2 + x + 1) = 0\). This gives a real root \(x = 1\) and a quadratic equation \(x^2 + x + 1 = 0\).

Step 3: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve \(x^2 + x + 1 = 0\), where \(a=1\), \(b=1\), \(c=1\).

Step 4: Find the Discriminant

Calculate the discriminant: \(b^2 - 4ac = 1^2 - 4 \times 1 \times 1 = 1 - 4 = -3\). The negative sign indicates complex roots.

Step 5: Calculate Roots Using Quadratic Formula

Substitute into the quadratic formula to find the roots: \(x = \frac{-1 \pm \sqrt{-3}}{2}\). This simplifies to \(x = \frac{-1 \pm i\sqrt{3}}{2}\).

Step 6: Identify the Cube Roots of Unity

The three cube roots of unity are 1, \(\frac{-1 + i\sqrt{3}}{2}\), and \(\frac{-1 - i\sqrt{3}}{2}\). These are the solutions to the original equation \(x^3 = 1\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. The imaginary part is typically represented as a multiple of the imaginary unit, denoted as \( i \), where \( i \) is defined by the property \( i^2 = -1 \). Thus, a complex number can be written in the form \( a + bi \), where \( a \) and \( b \) are real numbers.
Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane using the horizontal axis for the real part and the vertical axis for the imaginary part. This is what allows us to solve equations that have no real solutions, as we encounter cube roots of unity.

• Real Part: \( a \)
• Imaginary Part: \( b \)

Cube roots of unity provide a great example of using complex numbers, as their solutions include numbers with both real and imaginary parts.

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The formula for finding the solutions (or roots) of these equations is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates the values of \( x \) that satisfy the equation, providing solutions by considering the coefficients of the quadratic equation.

• It requires calculating the discriminant \( (b^2 - 4ac) \), which determines the nature of the roots.
• The solutions can be either real or complex numbers, depending on the value of the discriminant.

Understanding this formula is key when solving the quadratic part of polynomial factorization in cube roots of unity.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler polynomial factors that, when multiplied together, give the original polynomial. This technique simplifies solving polynomial equations.
For the cube roots of unity, we use this by rewriting the polynomial \( x^3 - 1 = 0 \) as a product of factors: \((x - 1)(x^2 + x + 1) = 0\).

• The factor \((x - 1)\) gives the real root: \( x = 1 \).
• The quadratic factor \(x^2 + x + 1\) leads us to solve a quadratic equation to find the remaining roots.

Effective factorization helps in isolating different components of the equations making it easier to solve them.

Discriminant

The discriminant in the quadratic formula \( b^2 - 4ac \) is a valuable indicator of the nature of the roots of a quadratic equation. This number tells us if the roots are real or complex, and if they are distinct or repeated.

• If the discriminant is greater than 0, the equation has two distinct real roots.
• If it equals 0, there are two identical real roots.
• If less than 0, the roots are complex and conjugate pairs.

In the context of the cube roots of unity, the discriminant for \( x^2 + x + 1 \) is -3. Since it's negative, this reveals the roots are complex, which aligns with our calculated roots involving the imaginary unit \( i \).

Roots of Unity

Roots of unity are special numbers that satisfy the equation \( x^n = 1 \) for a positive integer \( n \). They are evenly spaced around the unit circle in the complex plane and are fundamental in many areas of mathematics, including solving polynomial equations.
For \( n = 3 \), the polynomial equation \( x^3 = 1 \) has roots known as the cube roots of unity, which are 1, \( \frac{-1 + i\sqrt{3}}{2} \), and \( \frac{-1 - i\sqrt{3}}{2} \).

• The first root, 1, is real and represents the trivial solution where the power does not change the value of 1.
• The other two roots are complex conjugates, reflecting symmetry in their arrangement.

These roots are integral to understanding polynomial roots, transformations, and can offer insights into geometric interpretations within the complex plane.