Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Like Terms

The expression \((-6 + 6i) + (9 - i)\) consists of two parts. We need to identify and separate the real parts and the imaginary parts. In this expression, -6 and 9 are the real parts, while 6i and -i are the imaginary parts.

Step 2: Combine Real Parts

Add the real parts together: \(-6 + 9\). This simplifies to 3.

Step 3: Combine Imaginary Parts

Add the imaginary parts together: \(6i - i\). This simplifies to \(5i\).

Step 4: Write Result in Standard Form

Combine the results from Step 2 and Step 3 to express the final answer in the standard form \(a + bi\). This gives you \(3 + 5i\).

Key Concepts

Real and Imaginary Parts

Complex numbers are a fascinating concept in mathematics. They consist of two parts: a real part and an imaginary part. In any complex number denoted as \(a + bi\), \(a\) represents the real part, while \(bi\) represents the imaginary part. Here, \(i\) is the imaginary unit, with the property that \(i^2 = -1\). These two parts allow us to handle numbers and problems that are not solvable by using only real numbers.

Understanding the real and imaginary components is essential when performing operations on complex numbers. For instance, in the expression \((-6 + 6i) + (9 - i)\), we can see that \(-6\) and \(9\) are the real parts, and \(6i\) and \(-i\) are the imaginary parts. Recognizing these parts separates complex numbers into manageable units for mathematical operations.

Addition of Complex Numbers

The addition of complex numbers might look tricky at first, but it's quite straightforward once you understand the process.

When adding complex numbers, like in our expression \((-6 + 6i) + (9 - i)\), you need to add the real parts and the imaginary parts separately:

• Real Parts: \(-6 + 9 = 3\)
• Imaginary Parts: \(6i - i = 5i\)

By treating these parts independently, you simplify the addition process and avoid confusion. This method allows you to seamlessly incorporate each element of the complex numbers and culminates in a new complex number.

Standard Form of Complex Numbers

Every complex number can be expressed in its standard form \(a + bi\). This form is crucial as it provides a clear and consistent way to present complex numbers.

In our example, combining the real part \(3\) and the imaginary part \(5i\) gives us the complex number \(3 + 5i\). Presenting complex numbers in this format is essential for clarity, allowing anyone reading or using these numbers to immediately identify both the real and the imaginary components. This standard structure makes it easier for mathematicians and students alike to interpret results and perform further calculations with complex numbers.