Problem 22 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ -7 i+5 i $$

Step-by-Step Solution

Answer

The expression simplifies to \(-2i\).

Step 1: Identify Like Terms

Look at the expression \[-7i + 5i.\] Here, both terms are imaginary numbers and are like terms because they each have the imaginary unit "\(i\)."

Step 2: Combine Like Terms

To simplify the expression, combine the like terms by adding their coefficients: -7 and 5.Perform the addition: \(-7 + 5 = -2.\)

Step 3: Express the Result

Multiply the result by the imaginary unit \(i\):\(-2i.\)This simplified form is already the standard form of the given expression, where the real part is zero and the imaginary part is -2.

Key Concepts

Imaginary Unit

The imaginary unit is a mathematical concept used to extend the real number system. Represented by the letter \(i\), the imaginary unit is defined as the square root of -1. This means that \(i^2 = -1\). The introduction of the imaginary unit allows us to solve equations that do not have solutions within the set of real numbers, such as \(x^2 + 1 = 0\).
\(ewline\)Imaginary numbers are built upon this unit. Each imaginary number can be written in the form \(bi\), where \(b\) is a real number, indicating the coefficient of the imaginary unit. For example, in our exercise, terms like \(-7i\) and \(5i\) express different imaginary numbers. It’s crucial to understand that these numbers lie on the imaginary axis of the complex number plane- another dimension alongside the real number axis. This aids in mathematical calculations and problem solving involving complex numbers.

Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients. This concept is not limited to real numbers but also applies to complex numbers like those involving the imaginary unit \(i\).
\(ewline\)In our exercise, \(-7i\) and \(5i\) are like terms because they both involve the imaginary unit \(i\). When simplifying such terms, you only need to work with their coefficients. For instance, adding the coefficients \(-7\) and \(5\) gives \(-2\), allowing us to combine the terms into a single expression as \(-2i\).
\(ewline\)Remember, combining like terms is a fundamental skill in algebra that can help simplify expressions, ensuring they are easier to understand and work with in further calculations.

Standard Form

Standard form for complex numbers simplifies expressions to make them easy to interpret and work with. A complex number is expressed in standard form as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, which is multiplied by the imaginary unit \(i\).
\(ewline\)In our case, the expression simplifies to \(-2i\), which is already in standard form because it can be viewed as \(0 - 2i\). The real part \(a\) is \(0\) (since it's not visibly present) and the imaginary part \(b\) is \(-2\). This structure is important because it clearly separates the real and imaginary components, allowing for straightforward mathematical operations with complex numbers.
\(ewline\)Using standard form facilitates operations like addition, subtraction, multiplication, and division of complex numbers, making calculations more manageable.