Problem 31 - Intermediate Algebra

Question

Add or subtract. Write the sum or difference in the form \(a+b i .\) See Example 3 $$ 6-(8+4 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-2 - 4i\).

Step 1: Distribute the negative sign

Start by distributing the negative sign across the terms inside the parentheses. This means changing the sign of each term inside the parentheses:\[6 - (8 + 4i) = 6 - 8 - 4i\]

Step 2: Combine Real Parts

Combine the real numbers to simplify the expression. The real part of the expression is composed of 6 and -8:\[6 - 8 = -2\]

Step 3: Combine Imaginary Parts

Combine the imaginary parts of the expression. Here, we have only -4i:\[-4i\]

Step 4: Write in Standard Form

Now, write the sum or difference in the form \(a + bi\). Combining the results from Steps 2 and 3, we get:\[-2 - 4i\]

Key Concepts

Addition of Complex Numbers

Complex numbers are numbers that have both a real component and an imaginary component. When you're working with addition of complex numbers, it's a straightforward process - just add the real parts together, then add the imaginary parts together. This can be seen in the expression

• Adding two complex numbers, \( (a + bi) \) and \( (c + di) \), results in:\[(a+c) + (b+d)i\]
• Thus, only similar parts (real with real, imaginary with imaginary) are combined, emphasizing the importance of keeping components organized.

By lining up your real numbers separately from your imaginary numbers, you simplify the calculation and avoid mixing these two different "worlds". Addition of complex numbers maintains the same structure, just adjusted to include the added values of each component.

Subtraction of Complex Numbers

Subtracting complex numbers involves a similar process to addition, but with a twist - you have to carefully distribute the minus sign. This step is crucial to avoid mistakes, as the negative sign affects both parts within a set of parentheses.

• Suppose you want to subtract \( (c+di) \) from \( (a+bi) \):\[(a+bi) - (c+di) = (a-c) + (b-d)i\]
• Notice how each component is treated independently. In the subtraction example of \(6 - (8 + 4i)\), the real number \(6\) is simply subtracted by \(8\), while \(-4i\) stands alone once the operation is completed.

Using clear and organized steps ensures the operation is done accurately without confusing real and imaginary components.

Standard Form of Complex Numbers

The standard form of complex numbers is a neat and organized way to present them. It is expressed as \(a + bi\), where \(a\) is the real part, and \(b\) is the coefficient of the imaginary part \(i\), the imaginary unit.

• Easy identification of real and imaginary components in equations.
• Facilitates straightforward addition, subtraction, and comparison of complex numbers.
• Used in various mathematical and engineering contexts.

In the solution example, \(-2-4i\) is already in standard form. Recognizing this structure aids in calculation, communication, and offers a universal language for complex numbers.

Imaginary Numbers

Imaginary numbers extend our understanding of number systems, especially with the element \(i\), which is the square root of \(-1\). They enable solving equations that don't have real number solutions, broadening the possibilities of mathematical solutions.

• When squared, \(i\) yields \(-1\).
• Provides a foundation for complex numbers, allowing expressions like \(bi\) where \(b\) is a real number, to represent quantities involving \(\sqrt{-1}\).
• Very helpful in fields like engineering, physics, and signal processing, adding capabilities beyond real numbers alone.

Understanding imaginary numbers is fundamental as they underpin the operations and representations in the complex plane, revealing deeper layers of mathematics.