Problem 11 - Algebra and Trigonometry

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: Identify the real and imaginary parts

The expression \((2-5i) + (3+4i)\) consists of two complex numbers. The first complex number is \(2-5i\), where the real part is 2, and the imaginary part is -5i. The second complex number is \(3+4i\), where the real part is 3, and the imaginary part is 4i.

Step 2: Add the real parts together

Add the real parts of the two complex numbers together: 2 from the first number and 3 from the second number. \(2 + 3 = 5\).

Step 3: Add the imaginary parts together

Add the imaginary parts of the two complex numbers together: -5i from the first number and 4i from the second number. \(-5i + 4i = -1i\).

Step 4: Combine the results

Combine the results from Step 2 and Step 3. The real part is 5, and the imaginary part is -1. Therefore, the result is \(5 - 1i\).

Key Concepts

Real Part

When dealing with complex numbers, the real part is the component that is not multiplied by the imaginary unit, represented by the symbol "i". In mathematical terms, a complex number is often expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

• For example, in the complex number \(2 - 5i\), the real part is 2. This means that in a purely mathematical sense, if you were to disregard any parts multiplied by "i", this is what you are left with.
• Similarly, in the complex number \(3 + 4i\), the real part is 3.

Understanding the real part is crucial, especially when performing operations like addition on complex numbers, since each part must be handled separately to ensure accuracy.

Imaginary Part

The imaginary part of a complex number is the term that is multiplied by the imaginary unit "i", which is defined by the property \(i^2 = -1\). In a complex number expressed as \(a + bi\), \(bi\) is the imaginary part.

• For instance, in the complex number \(2 - 5i\), the imaginary part is \(-5i\). Note that the minus sign is included in this part.
• In the complex number \(3 + 4i\), the imaginary part is \(4i\).

When adding complex numbers, combining the imaginary parts separately from the real parts is crucial. This ensures that the operation respects both components of each number. Understanding imaginary parts helps in tasks such as simplifying expressions or interpreting the geometric implications of complex numbers.

Addition of Complex Numbers

Adding complex numbers involves combining their real and imaginary parts separately. Think of it as a step-by-step process where you're adding two layers of information. To add two complex numbers, like \((2-5i) + (3+4i)\), follow these steps:

• Step 1: Identify and add together the real parts. Here, you add 2 and 3, yielding 5.
• Step 2: Identify and add together the imaginary parts. Here, you add \(-5i\) and \(4i\), resulting in \(-1i\).
• Step 3: Combine the results from the previous two steps. The solution to the addition is \(5 - 1i\).

This operation demonstrates how complex numbers, although they have both a real and an imaginary component, can be manipulated systematically. Mastery of this concept aids in more advanced operations with complex numbers and is foundational in many fields of mathematics and engineering.