Problem 85 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((3-2 i)-(-2+5 i)\)

Step-by-Step Solution

Answer

The simplified expression in standard form is \(5 - 7i\).

Step 1: Identify the Components

The expression to simplify is \((3-2i) - (-2+5i)\). Identify the real parts: 3 and -2, and the imaginary parts: -2i and 5i.

Step 2: Distribute and Simplify

Distribute the negative sign through the second expression: \(3 - 2i + 2 - 5i\).

Step 3: Combine Real Parts

Add the real parts: \(3 + 2 = 5\).

Step 4: Combine Imaginary Parts

Add the imaginary parts: \(-2i - 5i = -7i\).

Step 5: Write in Standard Form

Combine the simplified real and imaginary parts to write the expression in the form \(a + bi\): \(5 - 7i\).

Key Concepts

Understanding Standard Form

The standard form of a complex number is represented as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The real part, \(a\), and the imaginary part, \(b\), allow us to express complex numbers in a clear and concise format. This form is essential for simplifying and understanding complex number expressions.

Writing complex numbers in standard form helps in comparing, adding, subtracting, and multiplying them with ease. For example, in this exercise, the final answer must be in \(a + bi\) format, ensuring clarity and consistency when performing operations on complex numbers. Knowing how to convert any complex format into its standard form is a crucial skill in mathematics.

Real and Imaginary Components Explained

Each complex number consists of two essential components: the real and the imaginary.

• Real Component: This is the non-imaginary part, represented as \(a\) in the standard form.
• Imaginary Component: This part is tied to the imaginary unit \(i\), represented as \(bi\). It involves the imaginary number \(i\), where \(i^2 = -1\).

Understanding these components helps to differentiate between the portions that are tied to real numbers and those connected with the imaginary unit.

In the given exercise, identifying these components was the first step of the solution. The real parts of the original expression \((3-2i)-(-2+5i)\) are 3 and -2. The imaginary components are -2i and 5i. Recognizing these parts is crucial for manipulating, simplifying, and ultimately writing the expression in standard form.

Applying the Distributive Property

The distributive property is a fundamental algebraic principle which states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b+c) = ab + ac\) holds true. This concept is essential when working with expressions, allowing for the simplification by distributing terms across addition or subtraction.

In the given problem, the distributive property is applied by redistributing the negative sign in the expression \( (3-2i)-(-2+5i) \). This step transforms the expression into \(3 - 2i + 2 - 5i\). By distributing, you essentially break down one larger expression into smaller, simpler components that can be easily combined or rearranged.

Applying this property is crucial for simplifying expressions down the line, especially when combining like terms to ultimately achieve the expression's standard form \(a + bi\). Understanding and mastering distribution will serve you well in various aspects of mathematics beyond just complex numbers.