Problem 24 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \left(\frac{2}{3}+12 i\right)\left(\frac{1}{6}+24 i\right) $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{-2593}{9} + 18i\).

Step 1: Distribute the Terms

First, apply the distributive property \((a+b)(c+d) = ac + ad + bc + bd\) to the expression \(\left(\frac{2}{3}+12i\right)\left(\frac{1}{6}+24i\right)\). This results in: \[ \left(\frac{2}{3}\right)\left(\frac{1}{6}\right) + \left(\frac{2}{3}\right)(24i) + (12i)\left(\frac{1}{6}\right) + (12i)(24i) \]

Step 2: Calculate Each Product

Calculate each of the products:1. \(\left(\frac{2}{3}\right)\left(\frac{1}{6}\right) = \frac{2}{18} = \frac{1}{9}\)2. \(\left(\frac{2}{3}\right)(24i) = \frac{48i}{3} = 16i\)3. \((12i)\left(\frac{1}{6}\right) = \frac{12i}{6} = 2i\)4. \((12i)(24i) = 12 \times 24 \times i^2 = 288i^2\)Since \(i^2 = -1\), \(288i^2 = -288\).

Step 3: Combine Like Terms

Now, combine the real and imaginary parts:- Real: \(\frac{1}{9} - 288 = -\left(288 - \frac{1}{9}\right) \)- Imaginary: \(16i + 2i = 18i\)Combine these results to express in the form \(a+bi\).

Step 4: Calculate Final Result

To express the numbers in the form of integers and fractions where needed, compute:\[-\left(288 - \frac{1}{9}\right) = -288 - \frac{1}{9} = \frac{-288 \times 9}{9} - \frac{1}{9} = \frac{-2592 - 1}{9} = \frac{-2593}{9}\]Therefore, the final complex number is \(\frac{-2593}{9} + 18i\).

Key Concepts

Distributive Property

The distributive property is a key principle in mathematics that allows us to multiply a single term across terms within a parenthesis. In algebra, this principle is expressed as \((a+b)(c+d) = ac + ad + bc + bd\). This property helps in simplifying expressions and solving equations.
In the context of complex numbers, where expressions often include imaginary parts like \(i\), the distributive property works the same way, just as it does with real numbers. The objective is to ensure each component interacts with others across the parenthesis, ensuring that none are left unmultiplied, capturing the entire equation correctly.

• First, multiply the real part of the first complex number by the real part of the second.
• Next, multiply the real part of the first complex number by the imaginary part of the second.
• Continue by multiplying the imaginary part of the first complex number by the real part of the second.
• Finally, multiply the imaginary part of the first with the imaginary part of the second, remembering that \(i^2 = -1\).

Once these individual products are calculated, they can be combined. This approach ensures a structured way to tackle problems involving complex expressions, offering clarity and precision.

Imaginary Unit

The imaginary unit, represented by \(i\), is a foundational concept in complex numbers. It is defined as the square root of \(-1\). This means \(i^2 = -1\), which defines how imaginary numbers interact with real numbers.
The notation of \(i\) allows us to solve problems that include square roots of negative numbers, which are not possible within the realm of real numbers. In the multiplication of complex numbers, as shown in this exercise, \(i\) follows the same arithmetic rules as any letter-like variable, but with the critical exception of its square:

• When \(i\) is multiplied by itself, remember that \(i^2\) results in a real number, specifically \(-1\).
• This conversion from imaginary to real components is crucial when simplifying expressions involving multiple terms with \(i\).

Using this property helps transform products of imaginary units into real values, simplifying complex calculations and allowing them to be expressed in the standard form.

Real and Imaginary Components

Complex numbers are expressed in the form \(a+bi\), where \(a\) is the real component and \(bi\) is the imaginary component. The real part \(a\) represents the non-imaginary portion of the number, while \(bi\) represents the multiples of the imaginary unit \(i\).
Understanding these components is essential because it lets us combine and manipulate numbers accurately:

• The real part deals with standard number operations and includes any component not involving \(i\).
• The imaginary part is strictly linked to values that include \(i\), allowing them to be isolated and simplified according to the properties of imaginary numbers.

In solving the exercise, separating these parts allows for easier computation. Real and imaginary components can be handled separately before being combined into the final solution. This form of separation aids in reducing complex arithmetic down to a more manageable form and highlights the importance of each part's unique role in complex number calculations.