Problem 87 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((4-2 i)+(9+4 i)\)

Step-by-Step Solution

Answer

The simplified expression is \(13 + 2i\).

Step 1: Identify like terms

We start by identifying the like terms in the complex numbers. In the expression \((4 - 2i) + (9 + 4i)\), the real parts are \(4\) and \(9\), and the imaginary parts are \(-2i\) and \(4i\).

Step 2: Add the real parts

We add the real parts of the complex numbers together: \(4 + 9\). This results in a sum of \(13\).

Step 3: Add the imaginary parts

Next, we add the imaginary parts of the complex numbers: \(-2i + 4i\). This simplifies to \(2i\).

Step 4: Write the expression in standard form

Combine the results from Step 2 and Step 3 to express the complex number in standard form: \(13 + 2i\).

Key Concepts

Standard Form

Complex numbers are often written in what's known as the "standard form." Essentially, the standard form of a complex number is expressed as \(a + bi\), where:\

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• \(a\) represents the real part.
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• \(b\) is the coefficient of the imaginary part (\(i\)).
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\The imaginary unit \(i\) has a fundamental property: \(i^2 = -1\). This makes complex numbers unique since they encompass both real and imaginary dimensions. \An important aspect of writing complex numbers in standard form is separating the real and imaginary components. This clear separation allows mathematicians and students alike to operate on them effectively and establish comparison. Whether dealing with simple operations or complex calculations, writing in standard form is crucial for clarity and communication.

Real and Imaginary Parts

Understanding what constitutes the real and imaginary parts of a complex number is foundational. In any complex number \(a + bi\), the real part is the number \(a\), and it behaves just like any other real number you might encounter in algebra.

The imaginary part is the term \(bi\). It's crucial to recognize that although it includes the real number \(b\), it represents a different dimension within the complex plane because of the presence of \(i\). This imaginary unit, \(i\), is not a real number but rather a tool to represent rotations and distance in this different dimension.

Recognizing these parts allows you to perform operations such as addition, subtraction, and even multiplication with ease. In any expression involving complex numbers, it becomes habitual to identify these parts explicitly, as we initially separate the components before engaging with the arithmetic operations.

Addition of Complex Numbers

Adding complex numbers follows rules similar to adding like terms in algebra. The core idea is to treat real and imaginary parts independently.

• To add two complex numbers, first add the real parts together.
• Next, add the imaginary parts.

For instance, consider the expression \((4 - 2i) + (9 + 4i)\). By separately adding the real parts, \(4\) and \(9\), you obtain \(13\). Similarly, add the imaginary parts: \(-2i + 4i\), which results in \(2i\).

By combining these results, the sum is \(13 + 2i\), expressed neatly in standard form. This method maintains the clarity of the expression while allowing straightforward simplification. Whether you're tackling textbook exercises or solving real-world problems, getting comfortable with the addition of complex numbers is a vital skill. It simplifies the process and ensures accuracy in results across various mathematical contexts.