Problem 29 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (2)(2+4 i) $$

Step-by-Step Solution

Answer

4 + 8i

Step 1: Distribute the real number

To write \((2)(2+4i)\) in standard form, start by applying the distributive property. This means we will multiply 2 by each part inside the parentheses.Multiply 2 by 2:\[2 \times 2 = 4\]Multiply 2 by \(4i\):\[2 \times 4i = 8i\]

Step 2: Combine and simplify

Now that each term has been distributed, combine the results from Step 1. The expression \(4 + 8i\) is already in its simplest form.The real part is 4, and the imaginary part is \(8i\).

Key Concepts

Standard Form in Complex Numbers

Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The standard form of a complex number highlights both its real and imaginary components, which makes it easier to perform arithmetic operations and understand the properties of the number.

• The real part, \( a \), represents the horizontal component on a complex plane.
• The imaginary part, \( bi \), represents the vertical component, where \( b \) is a real number, and \( i \) indicates the imaginary unit.

In our exercise, after simplification, we obtain \( 4 + 8i \) in standard form. Here, 4 is the real part, and \( 8i \) is the imaginary part. This format helps in clearly distinguishing the components of the complex number, and such clarity is crucial for both manual calculations and computer-generated solutions in mathematics.

Understanding the Distributive Property

The distributive property is a fundamental concept in mathematics, especially when dealing with algebraic expressions. It allows us to multiply a single term by each term within a set of parentheses independently in a structured manner. The property is expressed as:\[ a(b + c) = ab + ac \]Using the distributive property makes computations straightforward and minimizes errors. In our example, we applied the distributive property to the expression \( (2)(2 + 4i) \):

• First, multiply 2 (the outside term) by 2, producing 4.
• Next, multiply 2 by \( 4i \), resulting in \( 8i \).

This methodical distribution yields the expression \( 4 + 8i \) in standard form. Recognizing and utilizing the distributive property is essential for simplifying complex expressions accurately.

The Imaginary Unit and Its Importance

The imaginary unit, denoted as \( i \), is a unique symbol in mathematics with a special property: \( i^2 = -1 \). This concept extends the number system to include values that are not on the real number line.

• While real numbers have their place on a linear number system, complex numbers reside on a plane formed by the real axis and the imaginary axis.
• The imaginary unit helps in solving equations that have no real solutions, offering a complete set of solutions for polynomial equations.

In our exercise, the term \( 4i \) becomes \( 8i \) after applying the distributive property, showcasing how \( i \) operates within complex numbers. Understanding the imaginary unit is crucial when working in fields like electrical engineering, control systems, and anywhere complex numbers are used for modeling signals and waves.