Problem 50 - College Algebra with Modeling and Visualization

Question

Evaluate the expression with a calculator. $$ (-8.05-4.67 i)+(3.5+5.37) $$

Step-by-Step Solution

Answer

-4.55 + 0.70i

Step 1: Identify Real and Imaginary Parts

Observe that the expression is the sum of two complex numbers: (-8.05 - 4.67i) and (3.5 + 5.37). Notice the first term has a real part of -8.05 and an imaginary part of -4.67i. The second term has a real part of 3.5 and an imaginary part of 5.37i.

Step 2: Add the Real Parts

Add the real parts of the complex numbers: -8.05 (from the first number) and +3.5 (from the second number). Calculate: -8.05 + 3.5 = -4.55.

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers: -4.67i (from the first number) and +5.37i (from the second number). Calculate: -4.67 + 5.37 = 0.70.

Step 4: Combine the Results

Combine the sums from Steps 2 and 3 to form the resulting complex number: -4.55 + 0.70i.

Key Concepts

Real and Imaginary Parts

Complex numbers consist of a real part and an imaginary part. A complex number is typically represented in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part with \(b\) being a real number and \(i\) representing the square root of -1.

For example, consider the complex number \(-8.05 - 4.67i\). Here, the real part is \(-8.05\) and the imaginary part is \(-4.67i\). Similarly, for the complex number \(3.5 + 5.37i\), the real part is \(3.5\) and the imaginary part is \(5.37i\).

Identifying these parts is crucial. It helps you perform operations on complex numbers, such as addition, fairly straightforwardly.

Addition of Complex Numbers

Adding complex numbers is much like combining like terms in algebra. For any two complex numbers, say \(a + bi\) and \(c + di\), you add the real parts together and the imaginary parts together separately.

Consider the expression \((-8.05 - 4.67i) + (3.5 + 5.37i)\).

• First, add the real parts: \(-8.05 + 3.5 = -4.55\).
• Next, add the imaginary parts: \(-4.67 + 5.37 = 0.70\).

By adding these separately, you ensure each part retains its respective value, resulting in the final complex number \(-4.55 + 0.70i\).

This operation shows how complex numbers can be manipulated through straightforward arithmetic, simplifying otherwise complicated equations.

Mathematical Expressions Evaluation

Evaluating mathematical expressions involving complex numbers often requires breaking them down into simpler parts, like separating real and imaginary components.

In our example, we used a calculator to streamline the calculation steps for the expression \((-8.05 - 4.67i) + (3.5 + 5.37i)\). By directly adding the individual components, both real and imaginary, we simplify the problem to basic addition.

• First, simplify the real parts.
• Then simplify the imaginary parts.
• Finally, combine these results to obtain the full evaluated expression.

Using tools like calculators can further ensure accuracy, especially when dealing with decimals or more complex expressions. This method breaks down intimidating expressions into manageable tasks, making complex math much less daunting.