Problem 5 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ (5-9 i)+(2-4 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(7 - 13i\).

Step 1: Identify the Real and Imaginary Parts

In the given expression \((5-9i)+(2-4i)\), identify the real and imaginary parts of each term. The real part of the first term is 5, and the imaginary part is -9i. For the second term, the real part is 2, and the imaginary part is -4i.

Step 2: Add the Real Parts

Add the real parts of the two terms: \(5 + 2 = 7\). So, the real part of the result is 7.

Step 3: Add the Imaginary Parts

Add the imaginary parts of the two terms: \(-9i + (-4i) = -13i\). So, the imaginary part of the result is -13i.

Step 4: Write in the Form \(a+ib\)

Combine the calculated real and imaginary results to express the number in the form \(a+ib\). Thus, the entire expression simplifies to \(7 - 13i\).

Key Concepts

Addition of Complex Numbers

Complex numbers are a combination of a real part and an imaginary part. Adding complex numbers involves combining these parts separately. In the expression

• Real parts: add the real parts of the complex numbers together.
• Imaginary parts: add the imaginary parts of the complex numbers together.

For example, when we have a complex number addition like \[(5-9i)+(2-4i)\]we perform two separate addition operations. This simple two-step process ensures you accurately add complex numbers without missing any component, as each sums independently within its domain.

Real and Imaginary Parts

Every complex number has a real part and an imaginary part. These are the components that define the nature of the complex number.

• The real part is the component that does not involve the imaginary unit \(i\).
• The imaginary part is the component that involves \(i\), often written as a multiple of \(i\).

For instance, in \[x = 5 - 9i\]the real part is 5, and the imaginary part is -9. Recognizing these parts is crucial when performing operations like addition and subtraction, as each part interacts only with its counterpart from another complex number.

Complex Number Notation

The standard notation for complex numbers is in the form \(a + ib\), where \(a\) represents the real component and \(b\) identities the coefficient of the imaginary component. This form makes it clear how complex numbers are structured and makes the association with real and imaginary parts explicit. For any given complex number, this setup allows mathematicians to treat \(a\) as a regular real number and consider \(b\) as modifying the imaginary unit \(i\).
Using this notation aids both clarity and consistency, which is vital when dealing with multiple complex expressions and ensures everyone can follow the underlying structure of the numbers being used.

Basic Algebra

The basic principles of algebra apply to complex numbers just as they do to real numbers. Understanding these principles is essential when carrying out operations like addition and subtraction in complex numbers.

• Combine like terms: When adding complex numbers, always group real parts with real parts and imaginary parts with imaginary parts.
• Use of brackets: Particularly useful for keeping track of real and imaginary sections and ensuring proper operation order.

When working with an expression such as \[(5-9i)+(2-4i)\]applying basic algebraic rules solidifies the approach to outcome generation: sum each pair based on their type to find the answer. This method mirrors skills learned in early algebra, adapted to incorporate the imaginary unit.