Problem 62 - Algebra for College Students

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 \(54\).

Step 1: Recognize the Numbers

Identify the given numbers: We have two complex numbers, \(-6i\) and \(9i\), where \(i\) is the imaginary unit \(\sqrt{-1}\).

Step 2: Multiply the Imaginary Numbers

Multiply the numbers \((-6i)(9i) = -6 \times 9 \times i \times i = -54i^2\).

Step 3: Simplify Using Identity

Recall the identity of the imaginary unit: \(i^{2} = -1\). Thus, \(-54i^{2} = -54(-1)\).

Step 4: Final Result in Standard Form

Calculate the product using the identity: \(-54(-1) = 54\). Thus, the product in standard form is \(54 + 0i\).

Key Concepts

Understanding the Imaginary Unit

In the world of complex numbers, the imaginary unit is a fascinating concept. It's represented by the symbol \(i\). This imaginary unit is defined as \(\sqrt{-1}\). Since square roots of negative numbers are not real in the conventional sense, \(i\) opens up a new dimension in mathematics, providing a way to work with such numbers.

One of the most important properties of \(i\) is its ability to cycle through values when raised to increasing powers:

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

Once it reaches the fourth power, the cycle repeats itself. By understanding this cyclical nature, you can easily simplify expressions involving powers of \(i\). This property was used in our example to convert \(-54i^2\) into \(-54(-1)\), simplifying it further into a real number.

Standard Form of Complex Numbers

Complex numbers come in a special form known as the standard form. This is written as \(a + bi\), where \(a\) and \(b\) are real numbers. Here:

• \(a\) is the real part of the complex number.
• \(b\) is the imaginary part, which multiplies the imaginary unit \(i\).

For example, if we have the number \(54 + 0i\), it is expressed in standard form. The real part is \(54\), and the imaginary part, \(0\), indicates no imaginary component. Understanding the standard form is crucial as it allows you to easily add, subtract, and visualize complex numbers on the complex plane.

Multiplying Complex Numbers

When multiplying complex numbers, it's important to apply the distributive property. Here's how it works if we have two complex numbers, say \((a + bi)\) and \((c + di)\):

• Multiply each part separately: \(ac + adi + bci + bdi^2\).
• Combine and simplify: Remember \(i^2 = -1\), so \(bdi^2\) becomes \(-bd\).

In our exercise, the numbers given were purely imaginary, specifically \(-6i\) and \(9i\). This simplified our multiplication to a straightforward calculation:

• First, multiply the numerical coefficients: \(-6 \times 9 = -54\).
• Next, multiply the imaginary units: \(i \times i = i^2\).
• Simplify using the identity \(i^2 = -1\): hence \(-54i^2 = -54(-1) = 54\).

The final result, \(54\), is a real number, demonstrating how multiplication can change a complex number into a real one.