Problem 15 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$(2-5 i)+(3+4 i)$$

Step-by-Step Solution

Answer

The result is \(5 - i\).

Step 1: Understand the Problem

We are given the expression \((2-5i)+(3+4i)\) and need to find its sum. The answer should be expressed in the form \(a+bi\), where \(a\) and \(b\) are real numbers.

Step 2: Identify Real and Imaginary Parts

First, we identify the real parts and the imaginary parts of both complex numbers. The real parts are 2 and 3, while the imaginary parts are -5i and 4i.

Step 3: Combine Real Parts

Add the real components together: \(2 + 3 = 5\). This gives us the real part of the sum.

Step 4: Combine Imaginary Parts

Add the imaginary components: \(-5i + 4i = -i\). This gives us the imaginary part of the sum.

Step 5: Write in Standard Form

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

Key Concepts

Real and Imaginary Parts

Complex numbers consist of two main components: a real part and an imaginary part. When you look at a complex number written as \(a + bi\), \(a\) is the real part and \(bi\) is the imaginary part.
The real part, \(a\), is simply a real number. It behaves like any number you encounter on the number line. For example, in the complex number \(2 - 5i\), the real part is 2.
The imaginary part contains the imaginary unit \(i\), which is defined such that \(i^2 = -1\). This means \(bi\) is a number that behaves differently from regular real numbers. In \(2 - 5i\), the imaginary part is \(-5i\). Remember, always identify these parts separately before performing any operation on complex numbers, which will make solving problems easier.

Addition of Complex Numbers

Adding complex numbers is more straightforward than it might seem at first glance. The key is to treat the real and the imaginary parts separately.
Consider two complex numbers, \( (2 - 5i) \) and \( (3 + 4i) \). Here’s how you add them:

• First, add the real parts: \(2 + 3 = 5\).
• Next, add the imaginary parts: \(-5i + 4i = -i\).

The resulting complex number from these operations is \(5 - i\). By always aligning the like terms—real with real, imaginary with imaginary—you simplify the process. Complex number addition is both easy to perform and a fun way to explore numerical operations beyond the regular realm of numbers.

Standard Form of Complex Numbers

Writing complex numbers in standard form means expressing them as \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component.
After performing operations such as addition or multiplication, it’s vital to express your answer in this format for clarity and consistency. In our example, after adding \((2 - 5i)\) and \((3 + 4i)\), we got \(5 - i\) as the result.

• The number \(5\) is the real part \(a\).
• The term \(-i\) represents the imaginary part \(bi\).

Even if the imaginary part is negative or has a coefficient of 1, it should be written in its full form for proper understanding, as in \(5 + (-1)i\) which is simplified to \(5 - i\). This standard form maintains a uniform approach, making it easier for everyone to read and understand complex numbers.