Problem 29 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (2+4 i)+(6-5 i) $$

Step-by-Step Solution

Answer

The result is \(8 - i\).

Step 1: Identify the Real and Imaginary Parts

For the expression \((2 + 4i) + (6 - 5i)\), separate the real and imaginary parts. The real parts are 2 and 6, and the imaginary parts are \(4i\) and \(-5i\).

Step 2: Add the Real Parts

Add the real parts of the two complex numbers together. That is, \(2 + 6 = 8\).

Step 3: Add the Imaginary Parts

Add the imaginary parts of the two complex numbers. Remember to add the coefficients of \(i\): \(4 + (-5) = -1\). Thus, the imaginary part is \(-1i\).

Step 4: Write the Final Answer in the Form \(a + bi\)

Combine the results from Step 2 and Step 3 into a complex number format: The final expression is \(8 - 1i\), which is simply written as \(8 - i\).

Key Concepts

Real and Imaginary Parts

Complex numbers are composed of two parts: the real part and the imaginary part. These can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part multiplied by the imaginary unit \(i\). For example, in the complex number \(2 + 4i\), 2 is the real part and \(4i\) is the imaginary part.

Understanding the distinction between these parts is crucial because it allows us to perform arithmetic operations on complex numbers. Always separate the real and imaginary parts before proceeding with addition or subtraction tasks. This separation helps maintain clarity during calculations and ensures accuracy in your final answer.

Addition and Subtraction of Complex Numbers

When adding or subtracting complex numbers, treat real and imaginary parts separately. Consider two complex numbers: \((a + bi)\) and \((c + di)\).

• To add these numbers, combine their real parts: \(a + c\).
• Then, add their imaginary parts: \(b + d\).

This results in a new complex number: \((a+c) + (b+d)i\). For subtraction, follow the same principle but subtract the corresponding parts instead.

Let's look at an example: add \((2 + 4i)\) and \((6 - 5i)\).

• Real parts: \(2 + 6 = 8\).
• Imaginary parts: \(4 + (-5) = -1\).

Thus, the result is \(8 - i\). Always ensure your resulting complex number is in the form \(a + bi\). This approach simplifies dealing with complex numbers and makes the arithmetic process straightforward.

Imaginary Unit \(i\)

The imaginary unit \(i\) is a mathematical concept that helps us deal with square roots of negative numbers. The defining property of \(i\) is that \(i^2 = -1\). This property allows us to work with numbers that we cannot place on the traditional number line.

In the realm of complex numbers, \(i\) acts as a bridge between real and imaginary components. Consider it like a "marker" for the imaginary part. In any equation or expression involving complex numbers, \(i\) shows where the imaginary part starts. It is always accompanied by a real coefficient that tells how much of the imaginary unit is present, like \(4i\) in our example.

Always keep in mind the rule that \(i^2 = -1\). This is essential for simplifying expressions and ensures you can further manipulate complex equations effectively.