Problem 28 - Precalculus Mathematics for Calculus

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 equals \( -\frac{2581}{9} + \frac{162}{9}i \).

Step 1: Apply the distributive property

To multiply two complex numbers, use the distributive property (FOIL method). Multiply each term in the first complex number by each term in the second. In this case, distribute each term of \( \left( \frac{2}{3} + 12i \right) \) to \( \left( \frac{1}{6} + 24i \right) \).

Step 2: Multiply the real parts

First, multiply the real part of the first complex number by the real part of the second: \( \frac{2}{3} \times \frac{1}{6} = \frac{2}{18} = \frac{1}{9} \).

Step 3: Multiply the real part by the imaginary part

Next, multiply the real part of the first complex number by the imaginary part of the second: \( \frac{2}{3} \times 24i = \frac{48}{3}i = 16i \).

Step 4: Multiply the imaginary part by the real part

Now, multiply the imaginary part of the first complex number by the real part of the second: \( 12i \times \frac{1}{6} = 2i \).

Step 5: Multiply the imaginary parts

Multiply the imaginary parts together: \( 12i \times 24i = 288i^2 \). Since \( i^2 = -1 \), this becomes \( 288(-1) = -288 \).

Step 6: Combine all terms

Add up all these results to get the final complex number: \( \frac{1}{9} + 16i + 2i - 288 \). Simplify by combining like terms: \( -\frac{2581}{9} + 18i \).

Step 7: Convert each term to a common denominator

Rewrite \( -\frac{2581}{9} + 18i \) so both parts are shown with the denominator of 9. Convert 18i into a denominator to a form \( \frac{162}{9} i \).

Step 8: Final expression

The expression in the form \( a + bi \) is \( -\frac{2581}{9} + \frac{162}{9} i \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that is often used when multiplying numbers or expressions. Simply put, it allows you to "distribute" or spread a multiplication across an addition or subtraction. When dealing with complex numbers, the distributive property plays a crucial role in ensuring every part of one number is multiplied by every part of the other number.
In our original exercise, this means each term in the first complex number, \( \left( \frac{2}{3} + 12i \right) \), is multiplied by each term in the second complex number, \( \left( \frac{1}{6} + 24i \right) \).
This results in four separate multiplications that can then be combined to find the final result. Each step ensures that no part of the expression is left unaccounted for, allowing for an accurate calculation of the product.

FOIL Method

The FOIL method is a mnemonic that helps remember steps in the multiplication of two binomials. FOIL stands for First, Outer, Inner, Last which refers to the order of multiplying each of the terms together. This technique is particularly helpful when multiplying complex numbers.
Using the FOIL method on our expressions, the steps are:

• First: Multiply the first terms of each binomial (\( \frac{2}{3} \times \frac{1}{6} \)).
• Outer: Multiply the outer terms (\( \frac{2}{3} \times 24i \)).
• Inner: Multiply the inner terms (\( 12i \times \frac{1}{6} \)).
• Last: Multiply the last terms (\( 12i \times 24i \)).

By following these steps, you ensure every part of each binomial is considered in the multiplication, leading to a comprehensive solution.

Imaginary Unit

The imaginary unit, commonly denoted by \( i \), is a mathematical concept used to extend the real number system to include solutions to equations like \( x^2 + 1 = 0 \).
This is based on the premise that \( i^2 = -1 \).
In practice, the imaginary unit allows us to work with numbers that are not real, known as complex numbers which have both real and imaginary components (for instance, in the number \( a + bi \), \( a \) is the real part and \( b \) is the imaginary part).
In the step-by-step solution for our problem, we see the imaginary unit \( i \) used, particularly in the multiplication of imaginary parts. Notably, when the result involves \( i^2 \), it's crucial to remember it simplifies to \( -1 \), as seen when 288\( i^2 \) converts to \(-288\). This simplification is essential in arriving at the correct final solution.

Complex Multiplication

Complex multiplication involves combining two complex numbers that have both real and imaginary components. The process might seem tricky, but it closely follows principles outlined in the distributive property and FOIL method.
To multiply complex numbers, each part of the first is multiplied by each part of the second. Let's break it down further:

• Multiply real parts together.
• Multiply real parts by imaginary parts.
• Multiply imaginary parts together and convert \( i^2 \) to \(-1\) as necessary.

This may involve recalculating both real and imaginary parts, but when all terms are combined, you get a new complex number in the form \( a + bi \).
In our example, through careful calculation and combining each result, we transform the initial complex multiplication into a simplified form, \( -\frac{2581}{9} + \frac{162}{9} i \), showing the real and imaginary components clearly.