Problem 19 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (4-7 i)+(2+3 i) $$

Step-by-Step Solution

Answer

The result is \(6 - 4i\).

Step 1: Identify the Parts to Combine

In the expression \[(4-7i)+(2+3i),\]we need to combine like terms: the real numbers with each other and the imaginary numbers with each other.

Step 2: Add the Real Parts

The real parts of the expression are 4 and 2. Add them together:\[4 + 2 = 6.\]

Step 3: Add the Imaginary Parts

The imaginary parts of the expression are -7i and 3i. Add these together:\[-7i + 3i = -4i.\]

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to write the final answer in the form \(a + bi\):\[6 - 4i.\]

Key Concepts

Real Parts

In complex numbers, the real part is the component without the imaginary unit \(i\). Imagine a number like \(4 - 7i\). Here, the real part is \(4\). It sits before the imaginary piece and doesn't involve \(i\) at all.
If we look at another complex number, \(2+3i\), the real part is \(2\).
When adding or subtracting complex numbers, focus first on these real parts:

• Add the real components from both numbers (e.g., \(4 + 2\) becomes \(6\)).
• This step is just like adding regular numbers.

Understanding real parts helps simplify complex expressions by focusing on one piece at a time.

Imaginary Parts

The imaginary part is what makes complex numbers unique. It's expressed with \(i\), the square root of \(-1\). For the complex number \(4 - 7i\), the imaginary part is \(-7i\). It involves the special unit \(i\) that lets us work with the square root of negative numbers.
In another number, such as \(2+3i\), the imaginary part is \(3i\).
When dealing with these components, remember to:

• Combine only with other imaginary terms (e.g., \(-7i + 3i = -4i\)).
• The result remains an imaginary term, using \(i\).

Recognizing imaginary parts ensures that calculations acknowledge the distinct characteristics of the \(i\) term.

Addition of Complex Numbers

Adding complex numbers is straightforward once you break it down. Treat it like handling two separate issues. First, address the real parts, and then manage the imaginary parts.
Look at the expression \((4-7i)+(2+3i)\).
Think of the process as cleaning your room in zones:

• First, focus on the real numbers: Calculate \(4 + 2 = 6\).
• Then, turn to the imaginary numbers: Compute \(-7i + 3i = -4i\).

Finally, put both answers together to create the complete complex number. You end up with \(6 - 4i\). This method ensures that each component of complex numbers is handled correctly, preserving their properties in the overall expression.