Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Multiply or divide as indicated. Simplify each answer. $$(-2+3 i)-(-4+3 i)$$

Step-by-Step Solution

Answer

The simplified answer is 2.

Step 1: Distribute the Negative Sign

The expression given is \((-2+3i) - (-4+3i)\). First, distribute the negative sign across the terms in the second complex number. This changes \(-(-4 + 3i)\) to \(+4 - 3i\). Now, the expression becomes \(-2 + 3i + 4 - 3i\).

Step 2: Combine Like Terms

Next, combine the real parts and the imaginary parts separately. The real parts are \(-2\) and \(+4\), and the imaginary parts are \(+3i\) and \(-3i\). When you add the real parts: \(-2 + 4 = 2\). When you add the imaginary parts: \(3i - 3i = 0\).

Step 3: Write the Simplified Answer

Combine the results from Step 2 to write the simplified answer as a complex number. The result is \(2 + 0i\), which simplifies to \(2\) as there is no imaginary part.

Key Concepts

Distribute Negative Sign

When working with complex numbers, it's crucial to handle negative signs properly, especially in expressions involving subtraction. In our exercise, we start with \(-2 + 3i - (-4 + 3i)\). Here, the subtraction of a parenthesis term (-4 + 3i) requires distributing the negative sign across all terms inside the parenthesis.

• The negative sign changes the sign of each term: \(-(-4) = +4\) and \(-(3i) = -3i\).
• After distribution, the expression becomes \(-2 + 3i + 4 - 3i\).

This step is essential because failing to distribute the negative correctly leads to incorrect terms, throwing off the entire calculation.

Combine Like Terms

After distributing the negative sign, the next step is to combine like terms. In complex numbers, you'll typically have real and imaginary parts.

• Real parts are the plain numbers (without the \(i\) symbol). In this example, the real parts are \(-2\) and \(+4\).
• Imaginary parts are the numbers paired with \(i\). In this expression, they are \(+3i\) and \(-3i\).

To combine like terms, you sum the real parts together and the imaginary parts separately:
- For the real parts: \(-2 + 4 = 2\).
- For the imaginary parts: \(3i - 3i = 0\).
This separation is crucial because it ensures you accurately manage both parts of the complex number, which can otherwise lead to miscalculation if mixed.

Simplify Expression

After combining like terms, simplifying the expression is often the final step. For our particular exercise, the result after combining the terms is \(2 + 0i\). Simplification involves reducing this to its most concise form.

• Since the coefficient of \(i\) is zero, the imaginary part can be removed as it does not contribute to the value.
• Thus, \(2 + 0i\) simplifies simply to \(2\).

This final expression \(2\) is much cleaner and easier to work with. Always remember to strip away any terms that do not impact the final value of the expression, as it makes your answer clearer and more precise.