Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (2-8 i)-(2+8 i) $$

Step-by-Step Solution

Answer

The result is \(-16i\).

Step 1: Identify the complex numbers

The given expression is \[ (2-8i) - (2+8i) \].Here, we have two complex numbers: \[ 2 - 8i \] and \[ 2 + 8i \].

Step 2: Apply the subtraction rule

To subtract two complex numbers in the form \( a + bi \), we subtract the real parts and the imaginary parts separately.The real parts are: \( 2 \) and \( 2 \).The imaginary parts are: \(-8i \) and \(+8i \).

Step 3: Subtract the real parts

Subtract the real parts:\[ 2 - 2 = 0 \]

Step 4: Subtract the imaginary parts

Subtract the imaginary parts:\[ -8i - 8i = -16i \]

Step 5: Write the result in a+bi form

Combine these results to write the difference of the complex numbers:The real part is \( 0 \) and the imaginary part is \(-16i \), so the final answer is:\[ 0 - 16i \] or simply \[-16i \].

Key Concepts

Subtraction of Complex Numbers

Subtracting complex numbers is similar to how you would subtract polynomials. You handle the real and imaginary parts separately. To subtract two complex numbers, follow these steps:

• Identify the real and imaginary parts of each complex number.
• Subtract the real parts from one another.
• Subtract the imaginary parts from one another.

The key to subtraction is treating the imaginary part like another variable, such as \( x \) in algebra. This method ensures accuracy and simplicity.

Real and Imaginary Parts

Understanding complex numbers involves recognizing that each has a real and an imaginary part. The real part is the number without an 'i', and the imaginary part accompanies the 'i'.
For example, in the complex number \( 2 - 8i \), \( 2 \) is the real part, and \(-8i\) is the imaginary part.
By organizing these into real and imaginary, solving operations like addition or subtraction becomes straightforward and prevents confusion.

a + bi Form

The standard form for writing complex numbers is \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary component. This format is essential for consistency and understanding in mathematics.
When you perform operations such as addition or subtraction, the final result should be expressed in this form, making it clear which components are real and which are imaginary.

• In the subtraction of \( (2-8i) - (2+8i) \), the result \(-16i\) appears in the form \( 0 - 16i \), maintaining the \( a + bi \) structure even when the real part is zero.

Understanding the \( a + bi \) form helps anticipate how results should look during complex number operations.