Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (1+0 i)-\left(\frac{1}{5}-\frac{2}{5} i\right) $$

Step-by-Step Solution

Answer

The result is \(\frac{4}{5} + \frac{2}{5}i\).

Step 1: Break Down the Expression

The given expression to solve is \((1+0i) - \left(\frac{1}{5} - \frac{2}{5}i\right)\). It involves the subtraction of two complex numbers. Recall that when subtracting, the imaginary and real parts must be handled separately.

Step 2: Distribute the Negative Sign

Distribute the negative sign across \(\frac{1}{5} - \frac{2}{5}i\). This results in \(-\frac{1}{5} + \frac{2}{5}i\). The expression then becomes \((1 + 0i) + (-\frac{1}{5} + \frac{2}{5}i)\).

Step 3: Combine Real Parts

Add the real parts of the numbers separately: 1 and \(-\frac{1}{5}\). We get: \[1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}\]

Step 4: Combine Imaginary Parts

Next, add the imaginary parts: 0 and \(\frac{2}{5}\). Since the first imaginary part is 0, the result is simply \(0 + \frac{2}{5} = \frac{2}{5}i\).

Step 5: Form the Final Answer

Combine the results from Step 3 and Step 4. The sum in the form \(a + bi\) is \[\frac{4}{5} + \frac{2}{5}i\].

Key Concepts

Addition and Subtraction of Complex Numbers

Adding and subtracting complex numbers involves dealing with both the real and imaginary components separately. Consider two complex numbers: \(a + bi\) and \(c + di\). Whether adding or subtracting these, the real parts \(a\) and \(c\) are combined, while the imaginary parts \(b\) and \(d\) are combined separately.

For example, if we have \((1 + 0i) - \left(\frac{1}{5} - \frac{2}{5}i\right)\), we first distribute the negative sign across the second complex number, converting it to addition: \( (1 + 0i) + \left(-\frac{1}{5} + \frac{2}{5}i\right)\).

• Real parts: \(1 + \left(-\frac{1}{5}\right) = \frac{4}{5}\)
• Imaginary parts: \(0 + \frac{2}{5}i = \frac{2}{5}i\)

This results in the sum or difference in the form \(a + bi\), which is \(\frac{4}{5} + \frac{2}{5}i\) in this exercise.

a+bi Form

Complex numbers are expressed in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit which satisfies \(i^2 = -1\). This representation allows us to easily identify and manipulate the real and imaginary parts of complex numbers separately.

By packaging numbers in this form, operations like addition, subtraction, multiplication, and division are simplified, keeping the imaginary and real components clear and manageable. In our task, we subtracted complex numbers, modifying the expression into \(\frac{4}{5} + \frac{2}{5}i\). This simplicity helps us see the result instantly and avoid mistakes in calculation.

• The expression always starts with the real part \(a\).
• The imaginary part \(bi\) follows, clearly separated with a plus or minus sign.

Remember, maintaining the \(a + bi\) form is vital for clarity, especially in multi-step calculations.

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is crucial. The real part, \(a\), is the number without the imaginary unit \(i\). The imaginary part, \(b\), is the coefficient of \(i\). For example, in \(1 + 0i\), \(1\) is the real part and \(0i\) indicates no imaginary component.

In our given operation, \((1 + 0i) - (\frac{1}{5} - \frac{2}{5}i)\), we maintain the distinction:

• \(1\) is the real part of the first complex number.
• \(\frac{1}{5}\) is the real part of the second.
• \(0\) and \(-\frac{2}{5}\) are the imaginary parts respectively.

Combining these correctly, we get \(\frac{4}{5} + \frac{2}{5}i\), clearly depicting both parts in our final answer.
It's helpful to separate these components in your mind and on paper to prevent confusion during calculations, enabling you to work accurately with complex numbers.