Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (2-3 i)-(3+2 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(-1 - 5i\).

Step 1: Distribute Subtraction

When performing subtraction with complex numbers, distribute the negative sign across the second complex number. Change the expression from \( (2-3i)-(3+2i) \) to \( (2-3i) + (-3-2i) \).

Step 2: Combine Real Parts

Add or subtract the real parts of the complex numbers.The real parts are \( 2 \) and \(-3 \). Thus, \( 2 + (-3) = -1 \).

Step 3: Combine Imaginary Parts

Add or subtract the imaginary parts of the complex numbers.The imaginary parts are \(-3i\) and \(-2i\). Thus, \(-3 - 2 = -5i \).

Step 4: Simplify the Complex Number

Combine the results from the previous steps to express the simplified complex number. The simplified form of the complex number is \(-1 - 5i\).

Key Concepts

Subtraction of Complex Numbers

When we talk about subtracting complex numbers, it is essentially like subtraction in regular arithmetic, but with a twist because of the imaginary part. A complex number is made up of two parts: a real part and an imaginary part. For example, in the complex number \( a + bi \), \( a \) is the real part and \( bi \) is the imaginary part.
To subtract complex numbers, we follow a straightforward process:

• First, distribute the negative sign to the second complex number. This steps helps in effectively changing the subtraction into addition of the opposite.
• Next, subtract the real parts from each other and then subtract the imaginary parts.

For the given exercise \((2-3i) - (3+2i)\), this process involves changing the operation to addition of \((2-3i)\) and the negated \((-3-2i)\), effectively making it \((2-3i) + (-3-2i)\). This helps us clearly separate the real and imaginary components for easy simplification.

Simplification of Complex Numbers

Simplification of complex numbers involves combining like terms, meaning we need to deal with the real parts together and the imaginary parts separately. Once we have converted subtraction into addition of negative terms, simplification becomes much more manageable.
Let's break it down:

• First, add together the real numbers from each complex number. This gives you the real part of your simplified complex number.
• Then, add the imaginary numbers (the terms with \(i\)) together to get the imaginary part.

In our example, from \((2-3i) + (-3-2i)\), we combine the real parts by calculating \(2 + (-3) = -1\) and the imaginary parts by calculating \(-3i + (-2i) = -5i\). Therefore, the simplified version is \(-1 - 5i\).
Each part becomes explicit, making it easier to see the final form of the complex number.

Imaginary Unit

The imaginary unit \(i\) is a concept that helps us extend the number system to include non-real numbers. The imaginary unit is defined by the property that \(i^2 = -1\).
In the context of complex numbers, an expression that includes \(i\) represents values that are not captured by the traditional real number line.

• The imaginary unit is key to defining complex numbers, which take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit.
• In the subtractive process of complex numbers, \(i\) functions just like any other variable, but it is important to remember its unique property \(i^2 = -1\) if operations go beyond basic addition or subtraction.

Understanding the role of \(i\) is crucial for performing operations on and simplifying complex numbers. It allows both real and imaginary information to combine seamlessly into the elegant framework of complex numbers.