Problem 22 - Algebra for College Students

Question

Add or subtract as indicated. $$(-2-3 i)-(-4-14 i)$$

Step-by-Step Solution

Answer

The result is \(2 + 11i\).

Step 1: Distribute the Negative Sign

To begin, distribute the negative sign across the second complex number. This changes the expression from \((-2-3i) - (-4-14i)\) to \((-2 - 3i) + (4 + 14i)\).

Step 2: Combine Real Parts

Combine the real parts from both complex numbers. The real part of \((-2 - 3i)\) is \(-2\) and the real part of \((4 + 14i)\) is \(4\). Therefore, adding them will give:\(-2 + 4 = 2\).

Step 3: Combine Imaginary Parts

Combine the imaginary parts from both complex numbers. The imaginary part of \((-2 - 3i)\) is \(-3i\) and from \((4 + 14i)\) is \(14i\). Adding them results in:\(-3i + 14i = 11i\).

Step 4: Write the Final Solution

Combine the results from Step 2 and Step 3 to form the final complex number. The solution is:\(2 + 11i\).

Key Concepts

Addition of Complex Numbers

When adding complex numbers, combine the real parts and the imaginary parts separately. This operation is much like adding two binomials in algebra because each complex number contains two distinct components — one real and one imaginary. For instance, consider the complex numbers

• \((-2-3i)\) and
• \((4+14i)\).

Adding these involves two steps:

• First, add the real parts: \(-2\) and \(4\), which results in \(2\).
• Then, add the imaginary parts: \(-3i\) and \(14i\), resulting in \(11i\).

The sum of these two numbers is the complex number \(2 + 11i\). This showcases that the addition of complex numbers is straightforward and follows a systematic procedure, allowing easy calculations.

Subtraction of Complex Numbers

To subtract complex numbers, distribute the negative sign across the terms of the number being subtracted before combining like terms. This step is crucial because it involves changing the operation to addition of the opposite, simplifying the process.

• Take \((-2-3i) - (-4-14i)\) as an example.
• The first step is to change subtraction to addition: \((-2-3i)+(4+14i)\).
• Next, combine the real parts: \(-2 + 4 = 2\).
• Then, combine the imaginary parts: \(-3i + 14i = 11i\).

This results in the complex number \(2+11i\). It demonstrates that by converting subtraction into addition, complex number operations become more manageable.

Imaginary Numbers

Imaginary numbers are numbers that can be represented as a real number multiplied by the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). Imaginary numbers help us solve equations that don't have solutions in real numbers, such as \(x^2 + 1 = 0\). Here,

• \(-3i\)
• and \(14i\)

represent the imaginary components of our exercise. When working with imaginary parts, treat them as algebraic terms. Just like you add or subtract \(x\), you handle \(i\). With our example, subtracting the imaginary parts \(-3i\) from \(14i\) results in \(11i\). This process shows how imaginary numbers work in conjunction with real numbers to form complex numbers, allowing for comprehensive solutions.

Real Numbers

Real numbers are the opposite of imaginary numbers and can be found on the number line. They include whole numbers, fractions, and decimals, representing quantities that can be measured and counted. In our problem,

• \(-2\)
• and \(4\)

are real numbers.To perform operations on complex numbers, treat the real components as you would any typical arithmetic operation. In our example, subtracting \(-2\) from \(4\) yields \(2\), demonstrating how real numbers combine with imaginary numbers to form a new complex number. Real numbers provide the foundation upon which complex numbers are built, offering a robust structure for more advanced mathematics.