Problem 25 - Algebra for College Students

Question

Add or subtract as indicated. $$\left(-\frac{5}{9}+\frac{3}{5} i\right)-\left(\frac{4}{3}-\frac{1}{6} i\right)$$

Step-by-Step Solution

Answer

The result is \(-\frac{17}{9} + \frac{23}{30}i\).

Step 1: Understand the Expression

The expression given is the difference between two complex numbers, \[\left(-\frac{5}{9}+\frac{3}{5} i\right)-\left(\frac{4}{3}-\frac{1}{6}i\right)\]Our task is to subtract the second complex number from the first one, paying attention to both the real and imaginary components.

Step 2: Distribute the Negative Sign

Expand the subtraction by distributing the negative sign to both the real and imaginary parts of the second complex number:\[-\frac{5}{9} + \frac{3}{5}i - \frac{4}{3} + \frac{1}{6}i\]

Step 3: Combine Real Parts

Subtract the real parts of the complex numbers:\[R = -\frac{5}{9} - \frac{4}{3}\]Find a common denominator (9), then:\[R = -\frac{5}{9} - \frac{12}{9} = -\frac{17}{9}\]

Step 4: Combine Imaginary Parts

Subtract the imaginary parts of the complex numbers:\[I = \frac{3}{5}i + \frac{1}{6}i\]Find a common denominator (30), then:\[I = \frac{18}{30}i + \frac{5}{30}i = \frac{23}{30}i\]

Step 5: Form the Result

Combine the real and imaginary parts to form the resulting complex number:\[-\frac{17}{9} + \frac{23}{30}i\]

Key Concepts

Addition and Subtraction of Complex Numbers

Complex numbers have both a real part and an imaginary part. When we add or subtract them, each component must be handled separately. For the problem \(-\frac{5}{9}+\frac{3}{5}\,i - (\frac{4}{3}-\frac{1}{6}\,i)\), it involves subtracting the whole second complex number from the first.

To perform the subtraction, we first distribute the negative sign through the terms of the second complex number. This results in the expression \(-\frac{5}{9}+\frac{3}{5}\,i - \frac{4}{3} +\frac{1}{6}\,i\).

The next step is to group and operate separately on the real and imaginary parts.

• Real parts are \(-\frac{5}{9} - \frac{4}{3}\).
• Imaginary parts are \(\frac{3}{5}\,i + \frac{1}{6}\,i\).

After performing these operations, we get the resultant complex number.

Real and Imaginary Parts

Each complex number is composed of a real part and an imaginary part. In the expression \(-\frac{5}{9}+\frac{3}{5}\,i\) and \(\frac{4}{3}-\frac{1}{6}\,i\), real parts are \(-\frac{5}{9}\) and \(\frac{4}{3}\), while imaginary parts are \(\frac{3}{5}\) and \(-\frac{1}{6}\).

When dealing with complex numbers:

• The real part is the term without the imaginary unit \(i\).
• The imaginary part is the coefficient of \(i\).

During subtraction or addition of complex numbers, combining real components and imaginary components separately simplifies calculations to a large extent, allowing easier arithmetic operations.

Finding Common Denominators

Finding a common denominator is vital when dealing with fractions in complex numbers, to ensure subtraction or addition is accurate. This exercise includes fractions \(-\frac{5}{9}\) and \(\frac{4}{3}\) for the real parts, and \(\frac{3}{5}\) and \(\frac{1}{6}\) for the imaginary parts.

The steps are:

• For the real parts, \(\frac{4}{3}\) is rewritten as \(\frac{12}{9}\), aligning both fractions under a common denominator of 9.
• For the imaginary parts, \(\frac{3}{5}\) and \(\frac{1}{6}\) are converted to \(\frac{18}{30}\) and \(\frac{5}{30}\) after adjusting to a common denominator of 30.

After aligning the fractions to common denominators, the expression simplifies through standard fractional operations, helping in forming the resultant complex number.