Problem 55 - College Algebra

Question

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{4+i}{i}+\frac{3-4 i}{1-i} $$

Step-by-Step Solution

Answer

The simplified complex number is \(\frac{9}{2} - \frac{9i}{2}\).

Step 1: Simplify Each Fraction Separately

Firstly, let's simplify \(\frac{4+i}{i}\). Divide both the numerator and the denominator by \(i\): \(\frac{4+i}{i} = \frac{4}{i} + \frac{i}{i} = \frac{4}{i} + 1\). Since \(\frac{4}{i} = -4i\), the expression becomes \(-4i + 1\).

Step 2: Simplify Second Fraction

Now simplify \(\frac{3-4i}{1-i}\). To simplify this, multiply the numerator and denominator by the conjugate of the denominator: \(\frac{3-4i}{1-i} \times \frac{1+i}{1+i}\). This gives \(\frac{(3-4i)(1+i)}{(1-i)(1+i)}\).

Step 3: Multiply and Simplify

Expand the numerator: \((3-4i)(1+i) = 3 + 3i - 4i - 4i^2 = 3 - i + 4 = 7 - i\). The denominator: \((1-i)(1+i) = 1 + i - i - i^2 = 1 + 1 = 2\). Thus, the expression is \(\frac{7-i}{2} = \frac{7}{2} - \frac{i}{2}\).

Step 4: Combine the Results

Now we've simplified both fractions: \(-4i + 1\) and \(\frac{7}{2} - \frac{i}{2}\). Combine them: \(1 - 4i + \frac{7}{2} - \frac{i}{2}\).

Step 5: Simplify the Entire Expression

Combine real parts and imaginary parts separately: \(1 + \frac{7}{2} = \frac{2}{2} + \frac{7}{2} = \frac{9}{2}\), and for imaginary parts: \(-4i - \frac{i}{2} = -\frac{8i}{2} - \frac{i}{2} = -\frac{9i}{2}\). The final simplified complex number is \(\frac{9}{2} - \frac{9i}{2}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, especially when dealing with complex numbers. A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)). Simplifying expressions involves reducing them to their simplest form while maintaining all the original values.When simplifying complex expressions, the key is to manage both the real and imaginary parts separately. Here's a friendly approach to do so:

• Break down the expression into smaller parts.
• Simplify each part by breaking them down into real and imaginary components.
• Combine like terms, i.e., combine all real parts together and imaginary parts together.

By organizing complex expressions into these more understandable parts, we make it easier to tackle more complex algebraic problems.

Imaginary Numbers

Imaginary numbers are numbers that, when squared, give a negative result. The most basic imaginary number is \(i\), which is defined as \(\sqrt{-1}\). In algebra, imaginary numbers are often used in conjunction with real numbers to form complex numbers, which are written as \(a + bi\).An important concept when working with imaginary numbers is multiplying them by other terms. Since \(i^2 = -1\), anytime you see \(i\) multiplied by itself, it should be replaced by \(-1\). This allows you to simplify expressions involving \(i\) efficiently.For example, in our step-by-step solution, the fraction \(\frac{4+i}{i}\) had to be simplified. By recognizing that dividing by \(i\) is equivalent to multiplying by \(-i\), we were able to simplify it to \(-4i + 1\). This helps put the expression in a more familiar and manageable form.

Algebraic Fractions

In algebra, fractions where the numerator and the denominator are polynomials are called algebraic fractions. Simplifying these fractions, especially when dealing with complex numbers, requires a good grasp of algebraic operations.To simplify a complex algebraic fraction, several steps are usually involved:

• Identify the conjugate of the denominator, if necessary. The conjugate of a number \(a + bi\) is \(a - bi\).
• Multiply both the numerator and the denominator by the conjugate. This helps eliminate any imaginary parts in the denominator, making it real.
• Simplify both the numerator and the denominator by expanding and combining like terms.

For instance, while simplifying \(\frac{3-4i}{1-i}\), using the conjugate \(1+i\) reduced the imaginary component in the denominator, simplifying the overall fraction to \(\frac{7}{2} - \frac{i}{2}\). This method ensures that the fraction is expressed in its simplest, most understandable form.