Problem 61 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (5 i)(4 i) $$

Step-by-Step Solution

Answer

-20

Step 1: Multiply the Imaginary Coefficients

Multiply the numerical coefficients of the imaginary units. In this case, you will multiply 5 by 4 to get 20.

Step 2: Use the Property of Imaginary Numbers

Recall that the square of the imaginary unit i is -1. Therefore, when you multiply i by i (i.e., i^2), the result is -1.

Step 3: Combine the Results

Combine the results from the previous steps: \( (5i)(4i) = 20(i^2) = 20(-1) = -20 \)

Step 4: Write the Final Answer

The product \((5i)(4i)\) simplifies to \(-20\). Express the answer in standard form as \(-20 + 0i\) since the imaginary part is zero.

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating extension of the number system. At its core lies the imaginary unit, denoted as \( i \). This number, \( i \), is defined such that \( i^2 = -1 \). This definition might seem a bit peculiar, as squaring any real number traditionally results in a positive outcome, but it opens the door to a rich new realm of numbers.

Imaginary numbers can be thought of as numbers that exist on a different plane, perpendicular to the real number line. This abstraction allows mathematicians to solve equations that don't have solutions within the realm of real numbers. For example, the equation \( x^2 + 1 = 0 \) has no real solutions, because the square of any real number is at least zero, but it does have an imaginary solution: \( x = i \) and \( x = -i \).

When working with imaginary numbers, one can perform arithmetic operations like addition, subtraction, multiplication, and division. However, it's crucial to remember:

• The main property of \( i \), which is \( i^2 = -1 \).

Multiplication of Complex Numbers

Multiplying complex numbers, such as those in the given exercise, involves not just multiplying numbers but also considering the properties of \( i \). A complex number is generally expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

To multiply complex numbers:

• Multiply just like you would binomials, following the distributive property (often remembered as FOIL - First, Outer, Inner, Last).
• Combine like terms, specifically the imaginary terms, and apply the rule \( i^2 = -1 \).

In our specific case, with \( (5i)(4i) \):

• First, multiply the coefficients: \( 5 \times 4 = 20 \).
• Next, since both numbers have an \( i \), multiply them to get \( i^2 \) which equals \(-1\).
• Finally, combine these results: \( 20 imes (-1) = -20 \).

Standard Form of Complex Numbers

The standard form of a complex number is a somewhat structured way of representing these numbers, expressed as \( a + bi \), where \( a \) and \( b \) are real numbers.

In this form:

• \( a \) is the real part.
• \( bi \) is the imaginary part.

This notation allows for a clear distinction between real and imaginary components, facilitating operations and visual representation on a complex plane.

In solving the exercise, the product was simplified to \( -20 \). Since there was no remaining imaginary component after simplifying, it is expressed in standard form as \( -20 + 0i \). This makes it clear there's no imaginary part involved in the result. Even if the imaginary part is zero, it is often noted as \( 0i \) to maintain clarity and consistency in the notation.