Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$-\left(\frac{3}{2}+\frac{5}{2} i\right)+\left(\frac{5}{3}+\frac{11}{3} i\right)$$

Step-by-Step Solution

Answer

The result in standard form is \(\frac{1}{6}+\frac{7}{6}i\)

Step 1: Separate Real and Imaginary Parts

Begin by separating the real and imaginary parts of the complex numbers. We have -\(\left(\frac{3}{2}+\frac{5}{2} i\right)\) and +\(\left(\frac{5}{3} + \frac{11}{3} i\right)\). So, the real parts are -\(\frac{3}{2}\) and \(\frac{5}{3}\), while the imaginary parts are -\(\frac{5}{2}i\) and \(\frac{11}{3}i\).

Step 2: Add/Subtract Real Parts

Next step is the addition/subtraction of the real parts. Therefore, we individually add the real parts: -\(\frac{3}{2}\) + \(\frac{5}{3} = \frac{-9+10}{6}=\frac{1}{6}\).

Step 3: Add/Subtract Imaginary Parts

Next step involves the addition/subtraction of the imaginary parts: -\(\frac{5}{2}i + \frac{11}{3}i = -\frac{15}{6}i + \frac{22}{6}i = \frac{7}{6}i\).

Step 4: Combine Results

Finally, our complex number in standard form is the combination of the real and imaginary components that we calculated, which results in: \(\frac{1}{6}+\frac{7}{6}i\)

Key Concepts

Real Part

The real part of a complex number is the component that is not multiplied by the imaginary unit, \(i\). It behaves like a regular, real number that we are familiar with over our mathematical journey. When working with complex numbers, we split them into two parts:

• The real part
• The imaginary part

In our exercise, we start with two complex numbers: \(-\left(\frac{3}{2}+\frac{5}{2} i\right)\) and \(+\left(\frac{5}{3}+\frac{11}{3}i\right)\). Here, the real parts are \(-\frac{3}{2}\) and\(\frac{5}{3}\).
To find the resulting real part, we perform the operation:\(-\frac{3}{2} + \frac{5}{3}\).
This involves a simple subtraction/addition operation, where common fractions need to have a common denominator:

• Convert \(-\frac{3}{2}\) to \(-\frac{9}{6}\)
• Convert \(\frac{5}{3}\) to \(\frac{10}{6}\)
• Add/Subtract these values: \(-\frac{9}{6} + \frac{10}{6} = \frac{1}{6}\)

Providing the real component of the result.

Imaginary Part

The imaginary part of a complex number is accompanied by the imaginary unit \(i\), which represents the square root of -1. It forms the complex number when combined with the real part. For the complex number expression, \(-\left(\frac{3}{2}+\frac{5}{2} i\right)\) and \(+\left(\frac{5}{3} + \frac{11}{3} i\right)\), the imaginary parts are \(-\frac{5}{2}i\) and \(\frac{11}{3}i\).
To find the imaginary part of the resulting complex number, we need to handle the imaginary units:

• Convert \(-\frac{5}{2}i\) to \(-\frac{15}{6}i\)
• Convert \(\frac{11}{3}i\) to \(\frac{22}{6}i\)
• Add these imaginary components: \(-\frac{15}{6}i + \frac{22}{6}i = \frac{7}{6}i\)

This defines the imaginary component of the result.

Standard Form

The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This format provides a structured way to write complex numbers so that their components are immediately clear. In this exercise, we have calculated two components:

• Real part: \(\frac{1}{6}\)
• Imaginary part: \(\frac{7}{6}i\)

Thus, the final step is to tie these together into one expression. By combining these components, the complex number in standard form becomes:\(\frac{1}{6} + \frac{7}{6}i\).
This expression neatly organizes both the calculated real and imaginary parts into a familiar format, ready for further mathematical operations or interpretations.