Problem 62 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (-6 i)(9 i) $$

Step-by-Step Solution

Answer

The product is 54 + 0i.

Step 1: Multiply the Coefficients

Start by multiplying the numbers that are not associated with 'i', the imaginary unit. In this case, \[(-6) imes (9) = -54.\]

Step 2: Multiply the Imaginary Units

Next, multiply the imaginary units together: \[i imes i = i^2\]Remember that \(i^2\) equals \(-1\).

Step 3: Combine the Results

Combine the results from the previous steps:\[-54 imes i^2 = -54 imes (-1) = 54.\]

Step 4: Express in Standard Form

The resulting number from multiplying these complex numbers is a real number with no imaginary part. Therefore, the standard form of our result is \[54 + 0i.\]

Key Concepts

Imaginary Unit

When dealing with complex numbers, the imaginary unit is a crucial concept. It is represented by the symbol 'i' and is defined as the square root of -1. This means that:

• \(i = \sqrt{-1}\)

Understanding the imaginary unit allows us to handle numbers that are not real, known as imaginary numbers. An important property of 'i' is that when it is squared, the result is -1:

• \(i^2 = -1\).

This property is foundational in complex number arithmetic and is especially useful when multiplying complex numbers as we will see in the next sections.

Multiplication of Complex Numbers

Multiplying complex numbers involves more than just numbers alone. You have to take into account both the real and imaginary components. Let's quickly review how this works. Consider two arbitrary complex numbers in the form \(a + bi\) and \(c + di\). When you multiply them, you distribute the multiplication over the binomials:

• \((a + bi)(c + di) = ac + adi + bci + bdi^2\)

In this expression, \(i^2\) can be replaced with -1, turning it into (using the property already discussed):

• \(ac + adi + bci - bd\)

Combining the real parts \(ac - bd\) and the imaginary parts \((ad + bc)i\), gives us:

• \((ac - bd) + (ad + bc)i\)

This breakdown simplifies the complex multiplication process, allowing a clear understanding of how each component contributes to the final product.

Standard Form of a Complex Number

The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This form allows for easy addition, subtraction, multiplication, and division of complex numbers. In our exercise solution, we arrived at a product that was real after simplification, 54, with no imaginary part. Hence, we expressed it in the simplest standard form:

• \(54 + 0i\)

This format ensures that even when the imaginary part is zero, the number still adheres to the standard expression guidelines. Emphasizing this structure helps in identifying both real and imaginary components quickly.