Problem 7 - Precalculus

Question

Simplify each of the following. (a) \((5-6 i)+(9+2 i)\) (b) \((5-6 i)-(9+2 i)\)

Step-by-Step Solution

Answer

(a) 14 - 4i; (b) -4 - 8i.

Step 1: Identify Real and Imaginary Parts

For each expression, identify the real and imaginary parts. In (a), the real parts are 5 and 9, and the imaginary parts are -6i and 2i. In (b), the real parts are 5 and 9, and the imaginary parts are -6i and 2i.

Step 2: Apply Addition to Complex Numbers

For (a) Add the real parts:\( 5 + 9 = 14 \)Add the imaginary parts:\( -6i + 2i = -4i \)

Step 3: Combine Results for Addition

Combine the results from Step 2 to get the simplified form for (a): \( 14 - 4i \)

Step 4: Apply Subtraction to Complex Numbers

For (b) Subtract the real parts:\( 5 - 9 = -4 \)Subtract the imaginary parts:\( -6i - 2i = -8i \)

Step 5: Combine Results for Subtraction

Combine the results from Step 4 to get the simplified form for (b): \( -4 - 8i \)

Key Concepts

Real and Imaginary Parts

Complex numbers are numbers that have a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as the square root of -1, meaning \(i^2 = -1\). Understanding the distinction between real and imaginary parts is crucial for mathematical operations involving complex numbers.For example:

• The complex number \(5 - 6i\) has a real part of 5 and an imaginary part of \(-6i\).
• The complex number \(9 + 2i\) has a real part of 9 and an imaginary part of \(2i\).

Identifying these parts helps us perform operations like addition and subtraction efficiently. By breaking down complex numbers into their real and imaginary components, we can handle them similarly to simple variables in algebra.

Addition of Complex Numbers

Adding two complex numbers involves separately adding their real and imaginary parts. This process is analogous to adding binomials in algebra but with a specific breakdown into real and imaginary components.Here's how it works:

• Add the real parts: This is the sum of the real components of the complex numbers. For instance, \(5\) and \(9\) added give us \(14\).
• Add the imaginary parts: This is the sum of the imaginary components. For instance, \(-6i\) and \(2i\) result in \(-4i\).

The result is a new complex number that combines these sums. In our example, the addition of \((5 - 6i) + (9 + 2i)\) yields the simplified form \(14 - 4i\). This method ensures that both parts of the complex numbers are accurately combined to maintain correct mathematical relationships.

Subtraction of Complex Numbers

Subtraction of complex numbers is similar to addition but involves subtracting respective parts. Just like subtraction of regular numbers, care must be taken to ensure the order of operations is followed and negative signs are correctly managed.When subtracting:

• Subtract the real parts: For \(5\) and \(9\), \(5 - 9\) gives \(-4\).
• Subtract the imaginary parts: For \(-6i\) and \(2i\), \(-6i - 2i\) results in \(-8i\).

Combining these results, the subtraction \((5 - 6i) - (9 + 2i)\) results in the complex number \(-4 - 8i\). By keeping the components segregated during operations, we ensure clarity and correctness.

Simplification of Expressions

The process of simplifying expressions involving complex numbers is essentially the act of combining the computed real and imaginary components into a single expression. This is the final step after conducting either addition or subtraction.During simplification:

• Ensure no further simplification or reduction of terms is possible, as complex numbers are already in a simplified format when expressed as \(a + bi\).
• Double-check arithmetic operations to ensure no errors. Precision is especially crucial when dealing with signs and operations between terms.

For example, the expression \((5 - 6i) + (9 + 2i)\) simplifies to \(14 - 4i\), and \((5 - 6i) - (9 + 2i)\) simplifies to \(-4 - 8i\). In these expressions, both real and imaginary parts are simplified to their simplest form while maintaining the operation correctness across all steps.