Problem 16 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$(2+5 i)+(4-6 i)$$

Step-by-Step Solution

Answer

The expression simplifies to \(6 - i\).

Step 1: Understanding the Expression

We need to evaluate the expression \((2+5i) + (4-6i)\). This expression is in the standard form of \(a + bi\), where \(a\) and \(b\) are real numbers.

Step 2: Separate Real and Imaginary Components

Identify the real components \(2\) and \(4\), and the imaginary components \(5i\) and \(-6i\) from the expression \( (2+5i) + (4-6i) \).

Step 3: Add Real Parts

Add the real parts of the expression: \(2 + 4 = 6\).

Step 4: Add Imaginary Parts

Add the imaginary parts of the expression: \(5i + (-6i) = -1i\).

Step 5: Combine Results

Combine the results of the real and imaginary parts. Thus, the expression simplifies to \(6 + (-1)i\) or \(6 - i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a vital part of complex numbers. They emerge when we take the square root of a negative number, which is not possible using only real numbers. That's where the unit imaginary number, denoted as \(i\), comes in.

This special number is defined by its unique property, where \(i^2 = -1\). This property allows mathematicians to express solutions to equations that have no real number solutions. For example, \(\sqrt{-1}\) becomes \(i\). Imaginary numbers, when paired with real numbers, form what's known as complex numbers.

The value associated with the imaginary component in a complex number is often expressed as \(bi\) in the form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of \(i\). In the original problem of \((2+5i)+(4-6i)\), the imaginary components are \(5i\) and \(-6i\). By understanding and identifying these imaginary parts, we can manipulate and combine complex numbers effectively.

Addition of Complex Numbers

Adding complex numbers involves combining their real parts and imaginary parts separately. It follows the simple concept of like terms, where you treat the imaginary and real components as separate entities.

Let's review the process using our original example:

• Identify the real numbers: In \((2+5i)+(4-6i)\), the real parts are \(2\) and \(4\).
• Add the real parts: \(2 + 4 = 6\).
• Identify the imaginary numbers: From the same problem, the imaginary parts are \(5i\) and \(-6i\).
• Add the imaginary parts: \(5i + (-6i) = -1i\).

By individually summing the real and imaginary parts, the complex number addition is simplified to \(6 - i\). This systematic approach prevents errors and clarifies each part's role in the sum.

Standard Form of Complex Numbers

The standard form of complex numbers is expressed as \(a + bi\), where \(a\) represents the real part, and \(bi\) represents the imaginary part. This notation is essential because it provides a clear structure for expressing both components of a complex number.

In any given expression, like the original exercise \((2+5i)+(4-6i)\), we strive to organize the answer in this standard form. Our solution, after adding both the real and imaginary parts, result in \(6 - i\), which is already in the standard form, \(a + bi\).

This standardization makes it easier to perform operations on complex numbers, compare them, or input them into mathematical equations. It also simplifies understanding complex solutions by clearly distinguishing what is real and what is imaginary.