Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \(\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{3}\)

Step-by-Step Solution

Answer

The result after using the Binomial Theorem to expand and simplify \(\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{3}\) is \(-\frac{1}{2} - \frac{i}{2}\).

Step 1: Identify the components of the binomial

First, identify the components of our binomial, \(a\) and \(b\), and its power, \(n\). In the given problem, \(-1/2\) is our \(a\), \(\sqrt{3}/2 i\) is our \(b\), and 3 is our \(n\).

Step 2: Obtain binomial coefficients

Next, calculate the binomial coefficients. For \(n = 3\), these are 1, 3, 3, 1, as per Pascal's triangle.

Step 3: Apply the Binomial Theorem

Now apply the coefficients and substitute \(a\), \(b\), and \(n\) into the binomial theorem. Here's what we get: \[ \left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{3} = {3 \choose 0}\left(-\frac{1}{2}\right)^3\left(\frac{\sqrt{3}}{2} i\right)^0 + {3 \choose 1}\left(-\frac{1}{2}\right)^2\left(\frac{\sqrt{3}}{2} i\right)^1 + {3 \choose 2}\left(-\frac{1}{2}\right)^1\left(\frac{\sqrt{3}}{2} i\right)^2 + {3 \choose 3}\left(-\frac{1}{2}\right)^0\left(\frac{\sqrt{3}}{2} i\right)^3.\]

Step 4: Distribute and Simplify

Lastly, compute each of the terms and add the results together to simplify. After this step, we get: \[ -\frac{1}{8} - \frac{\sqrt{3}}{8} i + \frac{3\sqrt{3}}{8} + \frac{9}{8} i - \frac{3}{2} i - \frac{9\sqrt{3}}{8} - \frac{\sqrt{3}}{2} i - \frac{3}{8} = -\frac{1}{2} -\frac{i}{2} \]

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, especially in fields such as algebra and calculus. A complex number is defined as a number of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). This seemingly simple addition of an imaginary unit to the real number system allows for a much richer structure, enabling the solution of equations that would have no solution within the system of real numbers alone.

For example, the equation \( x^2 + 1 = 0 \) has no real number solution, as squaring any real number cannot produce a negative result. However, within the complex number system, this equation has two solutions: \( x = i \) and \( x = -i \). The complex plane is a two-dimensional plane where the horizontal axis represents the real part of the complex number and the vertical axis represents the imaginary part. In the case of the exercise, \( -1/2 + \sqrt{3}/2 i \), we have a complex number where \( -1/2 \) is the real part and \( \sqrt{3}/2 \) is the imaginary part, multiplied by \( i \).

Understanding complex numbers is critical for the binomial expansion of complex expressions, as it entails raising a binomial containing a complex number to a given power and simplifying.

Binomial Expansion

The Binomial Theorem provides a powerful shortcut for expanding expressions raised to a power, eliminating the tedious process of multiplying the expression by itself repeatedly. It states that the expansion of \( (a + b)^n \) is the sum of terms that take the form \( {n \choose k}a^{n-k}b^k \), where \( k \) goes from 0 to \( n \), and the binomial coefficients \( {n \choose k} \) can be found using Pascal's Triangle or by calculation.

In the context of complex numbers, such as \(\left(-1/2 + \sqrt{3}/2 i\right)^3\), we use the Binomial Theorem to expand the expression. Each term in the expansion corresponds to a specific combination of the components \( a \) and \( b \). It's essential to understand the properties of powers of \( i \) to simplify these terms effectively, as the powers of \( i \) cycle through \( 1, i, -1, \) and \( -i \).

In our exercise, calculating the binomial expansion gives us terms that include powers of \( i \). To simplify the result, we combine like terms and simplify using the properties of \( i \), which is a critical step to reach the simplified form of the expression, as seen in the exercise solution.

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers that provides a quick and easy way to derive the coefficients needed for the binomial expansion. Each row corresponds to the coefficients of the expanded form of \( (a + b)^n \), where \( n \) is the row number.

To use Pascal's Triangle for our binomial expansion exercise, we locate row three, which corresponds to the power of \( n = 3 \). This row, indexed starting from 0, reads 1, 3, 3, 1. These are the binomial coefficients that will multiply each term in the expansion of \( (a + b)^3 \).

Visualizing Pascal's Triangle

Imagine a triangle where each number is the sum of the two numbers directly above it in the previous row. Starting with a single 1 at the top, the rows of Pascal's Triangle build downward:

• 1
• 1    1
• 1    2    1
• 1    3    3    1
• ...

Each row provides the coefficients needed for the binomial expansion where the elements of the row correspond to the coefficients \( {n \choose k} \) in the expansion formula. The beauty of Pascal's Triangle lies in its simplicity; it allows for the straightforward determination of coefficients without the need for complex calculations. This visual and numerical pattern is an invaluable tool for anyone studying binomial expansion.