Problem 21 - Algebra for College Students

Question

Add or subtract as indicated. $$(-1-i)-(-2-4 i)$$

Step-by-Step Solution

Answer

The result is \(1 + 3i\).

Step 1: Understand the Expression

The given expression is \[ (-1-i) - (-2-4i) \] which involves subtracting two complex numbers: \(-1-i\) and \(-2-4i\). We are tasked with simplifying this expression by performing the subtraction.

Step 2: Distribute the Negative Sign

Subtracting one complex number from another requires distributing the negative sign to each component of the second complex number.Apply the distribution:\[ (-1-i) - (-2-4i) = (-1-i) + (2 + 4i) \] The negative signs of each component of \(-2-4i\) become positive.

Step 3: Add Real Parts Together

Now, add the real parts of the complex numbers:\[-1 + 2 = 1\] Thus, the real part of the result is 1.

Step 4: Add Imaginary Parts Together

Next, add the imaginary parts of the complex numbers:\[-i + 4i = 3i\] Thus, the imaginary part of the result is \(3i\).

Step 5: Combine the Results

Combine the real part and the imaginary part from Steps 3 and 4 into a single complex number:\[1 + 3i\] This is the simplified form of the original expression.

Key Concepts

Addition and Subtraction of Complex Numbers

When we talk about complex numbers, we often deal with both real and imaginary components. To add or subtract complex numbers, we treat these components separately. Here's how it works:

Complex numbers are usually written in the form of \(a + bi\), where \(a\) is the real part and \(b\) times \(i\) is the imaginary part. The imaginary unit \(i\) satisfies the equation \(i^2 = -1\).

To add or subtract complex numbers:

• Start by lining up the real and imaginary parts separately.
• Combine the real parts together.
• Then, combine the imaginary parts separately.

This process allows us to simplify expressions involving complex numbers effectively. For instance, in the expression we were given, distributing the negative sign helped convert a subtraction problem into an addition problem, which simplifies the calculation.

Imaginary Numbers

Imaginary numbers might sound like something out of a sci-fi novel, but they are a crucial aspect of mathematics, especially in the field of complex numbers. The foundation of imaginary numbers is the imaginary unit \(i\), defined by the property \(i^2 = -1\).

This is a strange concept at first because, in the real number system, no number squared gives a negative result. However, imaginary numbers extend our ability to solve equations that aren't possible with real numbers alone. They have practical applications in engineering, physics, and computer science. For example, in our exercise, the imaginary components \(-i\) and \(4i\) are added together to form the result \(3i\), demonstrating how they are handled similarly to real numbers within complex expressions.

Mathematical Expressions

Mathematical expressions are the building blocks of solving any problem, and understanding them is key to mastering mathematics. Expressions can involve numbers, variables, and operations like addition, subtraction, multiplication, and division.

For complex numbers, expressions can become a bit more challenging because they include both real and imaginary components. The complexity lies in correctly treating these parts when simplifying or solving such expressions.

In algebra, when you encounter expressions containing complex numbers, it's essential to remember the basic operations and properties of numbers. Always handle the real and imaginary parts distinctly, as shown in our example expression \((-1-i)-(-2-4 i)\). This expression shows the transformation stemming from algebraic distribution and combination of like terms, which leads to the simplification of complex expressions.