Problem 102 - 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 expanded form of \(\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{3}\) is \( - \frac{3\sqrt{3}}{4} i + \frac {1 - 3\sqrt{3}}{8} \).

Step 1: Apply the Binomial Theorem

We can write the expression \(\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{3}\) as \((a - bi)^{3}\), where \(a = \frac{1}{2}\) and \(b = \frac{\sqrt{3}}{2}\). Applying the Binomial theorem \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \), we get \((\frac{1}{2})^{3} - 3(\frac{1}{2})^{2}\frac{\sqrt{3}}{2} i + 3(\frac{1}{2})(\frac{\sqrt{3}}{2})^2 i^2 - (\frac{\sqrt{3}}{2})^{3} i^3 \).

Step 2: Simplify the Result

Simplify each term separately. Remember that \(i^{2} = -1\) and \(i^{3} = -i\). This gives \(\frac{1}{8} - \frac{3\sqrt{3}}{16} i - \frac{3\sqrt{3}}{8} - \frac{3\sqrt{3}}{16} i \). Which can be further simplified to \( - \frac{3\sqrt{3}}{4} i + \frac {1 - 3\sqrt{3}}{8} \). Combin the real parts and the imaginary parts to get the final result.

Step 3: Final Result

After simplifying the real and imaginary terms, we get the final result \( - \frac{3\sqrt{3}}{4} i + \frac {1 - 3\sqrt{3}}{8} \).

Key Concepts

Complex Numbers

A complex number is a number that combines a real part and an imaginary part. It's often written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This imaginary part includes the imaginary unit \( i \), which is defined as \( i = \sqrt{-1} \). Such numbers extend the concept of one-dimensional numbers like integers and fractions to a two-dimensional system.
Complex numbers allow for the solution of equations that don't have real solutions. For example, the equation \( x^2 + 1 = 0 \) can be solved using complex numbers as \( x = i \) and \( x = -i \). They provide a way to handle conditionswhere traditional real numbers fall short, making them invaluable in fields like engineering and physics.

• Real numbers: only real parts
• Pure imaginary numbers: only imaginary parts
• Complex numbers: both real and imaginary parts

Expansion

Expansion in mathematics usually refers to expressing a mathematical expression in an extended form.When dealing with complex numbers, it often involves expanding powers of binomials using the Binomial Theorem.The Binomial Theorem is helpful when expanding expressions like \((a - bi)^n\).In the original exercise, the expression \((\frac{1}{2} - \frac{\sqrt{3}}{2}i)^3\) is an expansion using this theorem:\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]. This formula allows us to break down the power into terms that can be more easily evaluated, using substitution for simplification later on.

• Determines coefficients in expansions
• Facilitates calculations with complex expressions
• Employed extensively in algebra and calculus

Simplification

Simplification in mathematics involves reducing an expression to its most concise form without changing its value. For expressions involving complex numbers, this might include combining like terms, or rationalizingthe expression. In the exercise, once each term of the expansion is calculated, the aim is to combine like terms (which are real and imaginary parts) and use the properties of \( i \) (like \( i^2 = -1 \) and \( i^3 = -i \)).This makes it easier to comprehend and further use the result in calculations.Simplification is crucial because:

• It makes expressions more manageable
• Reduces computational complexity
• Helps in understanding underlying mathematical relationships

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental mathematical concept that represents the square root of -1. It's the building block of complex numbers, and it's defined such that: \( i^2 = -1 \). Higher powers of \( i \) can be derived from this basic property:

• \( i^1 = i \)
• \( i^2 = -1 \)
• \( i^3 = -i \)
• \( i^4 = 1 \)

In the context of the original exercise, the imaginary unit helps in simplifying powers of binomials.Expressions such as \( i^3 \) are simplified to \( -i \), demonstrating how it simplifies calculations involving complex numbers. Understanding these properties can greatly help in manipulating and simplifying complex mathematical expressions.