Problem 17 - Precalculus Mathematics for Calculus

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 Components

Begin by identifying the real and imaginary parts of each complex number in the expression \((-6+6i) + (9-i)\). The first complex number \(-6+6i\) has a real part of -6 and an imaginary part of 6i. The second complex number \(9-i\) has a real part of 9 and an imaginary part of -i.

Step 2: Combine Real Parts

Add the real components of the two complex numbers. Real part of \(-6+6i\) is -6, and real part of \(9-i\) is 9. Calculate: \(-6 + 9 = 3\). The combined real part is 3.

Step 3: Combine Imaginary Parts

Next, add the imaginary parts of the two complex numbers. Imaginary part of \(-6+6i\) is 6i, and imaginary part of \(9-i\) is -i. Calculate: \(6i - i = 5i\). The combined imaginary part is 5i.

Step 4: Write the Result in Standard Form

Now that we have combined the real and imaginary parts, write the result as a single complex number in the form \(a+bi\). The real part is 3, and the imaginary part is 5i.Thus, the expression simplifies to: \(3 + 5i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers involves a straightforward process, similar to adding vectors. Each complex number is composed of a real part and an imaginary part. For example, consider the complex numbers

• \(-6 + 6i\), where - the real part is \(-6\) - the imaginary part is \(6i\)
• \(9 - i\), where - the real part is \(9\) - the imaginary part is \(-i\)

When adding, you combine like terms:

• Real parts with real parts: \(-6 + 9 = 3\)
• Imaginary parts with imaginary parts: \(6i - i = 5i\)

Thus, the sum of these complex numbers is \(3 + 5i\). Always remember to align real parts with real and imaginary with imaginary, helping ensure accuracy in your calculations.

Real and Imaginary Parts

Understanding real and imaginary parts is essential when dealing with complex numbers. Each complex number is generally expressed as \(a + bi\), where - \(a\) is the real part- \(bi\) is the imaginary part.
The real part is the component without the imaginary unit \(i\), and it behaves much like a typical real number. The imaginary part includes the imaginary unit \(i\), which represents the square root of \(-1\).
Often in exercises, like the original one, you're tasked to identify and manipulate these parts separately before combining them into the final result. Such distinction simplifies operations like addition and subtraction, ensuring clarity in each step of the calculation process.

Standard Form of Complex Numbers

The standard form of a complex number is \(a + bi\), where both \(a\) and \(b\) are real numbers. The symbol \(i\) denotes the imaginary component. This form is crucial as it provides a consistent method to represent complex numbers, making both their interpretation and arithmetic operations more accessible.
In this form:

• The real number \(a\) appears first, explicitly showing the real portion of the complex number.
• The imaginary number \(bi\) comes second, indicating the imaginary part.

This arrangement ensures that complex numbers can be added, subtracted, or otherwise manipulated without confusion. The solution to expressions like \((-6 + 6i) + (9 - i)\) requires transforming the result into this form, yielding \(3 + 5i\) as the final answered representation.