Problem 13 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (-6+6 i)+(9-i) $$

Step-by-Step Solution

Answer

The result is \(3 + 5i\).

Step 1: Identify Real and Imaginary Parts

First, identify and separately group the real and imaginary parts of the complex numbers. In the expression \((-6 + 6i) + (9 - i)\), the real parts are \(-6\) and \(+9\). The imaginary parts are \(+6i\) and \(-i\).

Step 2: Add the Real Parts

Add the real parts of both complex numbers. Calculate: \(-6 + 9 = 3\).

Step 3: Add the Imaginary Parts

Add the imaginary parts of both complex numbers. Calculate: \(6i + (-i) = 5i\).

Step 4: Combine Real and Imaginary Parts

Combine the results from Step 2 and Step 3 to form the final complex number. Combine: \(3 + 5i\).

Key Concepts

Real and Imaginary Parts

Complex numbers have two fundamental components: the real part and the imaginary part. A complex number is typically expressed in the form \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part.

For example, in the expression \(-6 + 6i\), the real part is \(-6\), and the imaginary part is \(6i\). Similarly, for the complex number \(9 - i\), the real part is \(9\), and the imaginary part is \(-i\).

This distinction is crucial because it allows us to operate separately on the real and imaginary components when performing mathematical operations. By clearly identifying these parts, complex number calculations can be done efficiently, ensuring better understanding and solution accuracy.

Addition of Complex Numbers

To add complex numbers, you need to consider their real and imaginary parts separately. This method ensures clarity and accuracy in complex number arithmetic.

Here's how you should perform this operation:

• **Add the real parts**: Start by adding the real numbers from each complex number. For instance, in our given expression, add \(-6\) and \(9\), resulting in \(3\).
• **Add the imaginary parts**: Next, focus on the imaginary parts. Combine \(6i\) with \(-i\). This calculation yields \(5i\).

By handling real and imaginary parts individually, the process remains structured and straightforward. This method is a fundamental tool not only for simple operations but also for more advanced manipulations involving complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This form is crucial for readability and further mathematical operations.

After performing the operations of addition, like in our example, always ensure your result is in this standard form to maintain consistency and clarity. After combining the real and imaginary components separately, the new complex number is expressed in the form \(3 + 5i\).

Using this format is important because it aligns with typical mathematical conventions, facilitating easier calculations, comparisons, and interpretations of complex numbers in various applications.