Problem 21 - Precalculus

Question

Perform the indicated operation and express the result as a simplified complex number. \((-4+4 i)-(-6+9 i)\)

Step-by-Step Solution

Answer

The simplified complex number is \(2 - 5i\).

Step 1: Identify the elements

We need to subtract \((-6 + 9i)\) from \((-4 + 4i)\). This involves separately handling the real and the imaginary components of both complex numbers.

Step 2: Separate into components

For the complex number \((-4 + 4i)\), the real part is \(-4\) and the imaginary part is \(4i\). For \((-6 + 9i)\), the real part is \(-6\) and the imaginary part is \(9i\).

Step 3: Subtract the real parts

Subtract the real part of the second complex number from the real part of the first: \(-4 - (-6) = -4 + 6 = 2\).

Step 4: Subtract the imaginary parts

Subtract the imaginary part of the second complex number from the imaginary part of the first: \(4i - 9i = -5i\).

Step 5: Combine the results

Combine the results by writing the new real and imaginary parts together as a simplified complex number: \(2 - 5i\).

Key Concepts

Subtraction of Complex Numbers

When subtracting complex numbers, you need to think of them as simply pairs of numbers – much like coordinates on a graph. Each complex number consists of a real part and an imaginary part, typically expressed in the form \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component. To subtract one complex number from another:

• Subtract the real parts from each other.
• Subtract the imaginary parts from each other.

In our example of \((-4 + 4i) - (-6 + 9i),\)the process involves calculating:

• Real parts: \(-4 - (-6) = -4 + 6\)
• Imaginary parts: \(4i - 9i\)

This systematic approach helps manage the calculations clearly and concisely.

Real and Imaginary Components

Understanding the distinction between the real and imaginary components of a complex number is crucial in performing operations like addition or subtraction. A complex number \(a + bi\) has:

• a real component \(a\), which is the part of the number that we would be consider 'normal' numbers like 1, -3, or 5.5.
• an imaginary component \(bi\), where \(i\) is the imaginary unit defined as \(i^2 = -1\).

In the subtraction of our two complex numbers, we clearly identify each component:

• For \((-4 + 4i)\), the real part is \(-4\) and the imaginary part is \(4i\).
• For \((-6 + 9i)\), the real part is \(-6\) and the imaginary part is \(9i\).

This separation helps in precisely targeting which parts to combine or subtract during the operation.

Simplification of Complex Expressions

Simplification of complex expressions involves combining like terms to make the expression as concise as possible. Once the subtraction occurs, it's important to check that each part of the expression is simplified accordingly. In the example, you subtract the real parts and the imaginary parts separately:

• The real subtraction gives you \(2\).
• The imaginary subtraction provides \(-5i\).

Finally, you combine these results to express them in the standard complex number form, giving \(2 - 5i\). By organizing terms logically, the resultant expression is not only easier to interpret but also maintains mathematical accuracy, making it simpler for further calculations or analysis.